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This book should be returned on or before the date last marked below. 

Mathematics in Retrospect 

Studies in the Evolution of Mathematical Thought and Technique 

Written for those who would cultivate Mathematics either 

as a Vocation or as an Avocation 



by Tobias Dantzig 






The Bequest of 
the Greeks 





This book is copyright under the Berne Convention. Apart from 

any fair dealing for the purposes of private study, research, 

criticism or review, as permitted under the Copyright Act 1911, 

no portion may be reproduced by any process without written 

permission. Enquiry should be made to the publisher. 

George Allen & Unwin Ltd., 1955 

Printed in Great Britain 
in 1 1 point Baskerville 




The Bequest of the Greeks is a study of problems, principles 
and procedures which modern mathematics has inherited 
from Greek antiquity. Not all the ideas and issues which had 
agitated the great Greek minds, from Thales to Pappus, were 
destined to live. Some were still-born, others moribund, many 
withered on the vine, many more perished in the storm which 
all but obliterated the glory of Hellas. Much has been written 
on these topics, and this writer recognizes how important such 
studies in mathematical archaeology are to a comprehensive 
history of the field. However, The Bequest of the Greeks is not 
a history of Greek mathematics. It deals only with such issues 
as have survived the Grand Catastrophe, survived the long 
hibernation of the Dark Ages, survived even the growing 
pains of the Era of Restoration, and are alive today. 

This work grew out of the author's experience with another 
book, Number^ the Language of Science. Like that earlier work, 
the present volume is addressed to two categories of readers. 
Typical of the first group is the individual who has neither 
the preparation nor the taste for the technical aspects of 
mathematics, but who, upon reaching intellectual maturity, 
has come to realize its importance to contemporary thought 
and life. The gap in the mathematical education of such 
readers is more than offset by their eagerness to learn and 
their capacity of appraising and absorbing ideas. 

On the other hand, there is the ever-growing group of 
people who have acquired a more professional attitude towards 
mathematics. The typical individual in this category is, 
temperamentally, interested more in the "how" than in the 
"why" of things. Thus, the origin and the evolution of the 
methods he uses in his daily work should be of real concern to 
him, inasmuch as this will help him to appraise the validity 
of these methods as well as their limitations. But this is not 
all. Some of the practices of the Greek mathematicians have 
passed into oblivion not so much because modern discoveries 
have rendered them obsolete, but because they were lost in 


"the shuffle of history." Indeed, quite a few of these practices 
excel, both in efficacy and elegance, the routines which the 
individual has learned on the school bench. 

This desire on the part of the writer to reach two groups of 
readers, so dissimilar in attitude, taste and interest, accounts 
for the dual structure of the present work. Part One, The Stage 
and the Cast, is designed for the general reader, and no effort 
has been spared to stay within his mathematical ken. It is the 
author's belief that the average high school curriculum is 
fully adequate for this part of the work. Part Two, Anthology 
of the Bequest, is more technical in character. Indeed, most of 
the problems treated in the Anthology have only in recent 
times reached a stage of fruition, while some remain unsolved 
today. The reader with a penchant for mathematics will 
experience no difficulty with the principles implied; but a 
deeper understanding of these problems can be attained only 
through diligent exertion. In the words of Spinoza: "Omnia 
praeclara tam difficula quam rara sunt." All that is excellent 
is as difficult as it is rare. 

The Bequest of the Greeks is the first of three volumes which 
will appear under the collective title Mathematics in Retrospect. 
The second volume, Centuries of Surge, will follow shortly: its 
thesis is the rebirth of mathematics and its prodigious progress 
in the seventeenth and eighteenth centuries. The concluding 
volume of the trilogy, The Age of Discretion, which deals with 
the development of mathematics in the nineteenth century, 
is in preparation, but the writing has not yet reached a stage 
when a definite date can be set. 

The rest of this preface is the defence of a title. To describe 
in a few words the contents of a work is a difficult task at best, 
but when these words are to convey to the prospective reader 
the aims of the writer and the scope of his undertaking, then 
the task becomes formidable indeed. Yet, sooner or later, the 
writer of any book must face the problem of naming it. He 
passes in review a number of possible word-arrays, rejecting 
some because they err by excess, others because they err by 
default. In the end, he settles on a compromise, consoling 
himself that the preface may clear up the ambiguities of the 
chosen title. 



The title Mathematics in Retrospect is not only ambiguous, it 
is ambitious as well. Indeed, the term "mathematics" embraces 
so vast a field of human knowledge that only the writers of an 
encyclopaedia would be entitled to use it as a title of their 
work, and the present work is not an encyclopaedia in any 
sense of the word. To be sure, the centrifugal trend of the 
author's mind did cause his interests to scatter in many direc- 
tions; still, he found the more subtle aspects of modern mathe- 
matics beyond his ken or reach, with the result that such 
topics as topology, point sets, lattices, and many others too 
numerous to mention, will not be even touched upon. If, 
despite these gaps, the author resolved to use the title * 'mathe- 
matics" in all its bare audacity, it is because he could not 
find a more proper term to qualify the variety of topics handled 
in the book. 

The phrase in retrospect is worse than ambiguous : it is suscep- 
tible of at least three interpretations. While the words suggest an 
historical approach, they could fittingly adorn the title page 
of a curricular survey, and just as fittingly be used to describe 
a book of reminiscences. Strangely enough, any of these seem- 
ingly contradictory interpretations could aptly apply to the 
present work. 

For, though this work is not a history of mathematics, it 
does aim at restoring the historical perspective which the 
undergraduate curriculum has a tendency to distort. Thus, the 
historical approach has been freely used throughout in tracing 
mathematical ideas and processes to their sources and stressing, 
at the same time, the methods used by the masters of the 

Again, while this book is not a review of an undergraduate 
curriculum, it does aim at integrating the experience of an 
individual who, after being exposed to these ideas and pro- 
cesses during a protracted but rather immature period of 
life, is about to plunge into the more advanced realms of 
mathematics incident to graduate study. 

Finally, while this book is not a collection of reminiscences, 
the author has made no attempt to restrain the subjective 
element. Indeed, in a certain sense, this is the record of a man 
who for more than four decades has been studying mathe- 


matics, teaching it, writing it, and applying it. Rightly or 
wrongly, the author cherishes the hope that this record will 
prove to be of some value to those who aspire to cultivate 
mathematics either as a vocation or as an avocation. 


Pacific Palisades 
April, 1954 


The author wishes to express his indebtedness to the 
Carnegie Corporation of New York for its generous assistance 
during the critical period in the preparation of this work. 






Part One 

1 On Greeks and Grecians 15 

2 The Founders 20 

3 On the Genesis of Geometry 34 

4 Pyramids 46 

5 Pentacles 55 

6 The Pseudomath 72 

7 The Interdiction 84 

Part Two 

an Anthology of the Greek Bequest 

8 The Hypotenuse Theorem 95 

9 Triples 108 

10 The Crescents of Hippocrates 121 

11 The Quadratrix of Hippias 138 

12 The Algorithm of Euclid 145 

13 An Archimedean Approximation 152 

14 The Formula of Hero 160 

15 The Chords of Hipparchus 169 


INDEX 189 


Figure 1 

Pappus, (300AD) 
Mentions, (100 A$ 

Hero, too) 
Theon, (130) 
Diodes, U30) 
Hipparchus, 060) 
Apollonius, food @\ 
Erotasthene,Z76-M j 
Aristarchus,3/0230 @ 
Euclid, (300) 

Miletus, Ionia 
Samos, " 
(D Chios, 

Athens, Greece 
0) Delos, 


Crotona. Italy 
Syracuse, " 

Blea , 

Euclid to Hero (300B.C- B.C.) 

Aristotle, 384-3*t 
Dinostratus, (360) 


EUdOXUS, 408-355 
Plato, 427-347 

Demqcntu3 t 46O370 

Pythagoras, ssosoo 

Tha^S, 624-546 


@Abdera. Thrace 
Alexandria, Egypt 

QMaareth. Palestine 
Tire, Phoenicia 
(^Babylonia, Chaldea 

, Pamphitfa 


@Nicaea, Bithinia 
Carthage, Africa 

Thales to Euclid (600S.C-300B.C) 

Chapter One 


A dazzling light, a fearful storm, then unpenetrable 


The stage on which were enacted the early episodes of the 

drama which I am about to unfold was Ancient Greece; the 

cast bore names unmistakably Greek; and the medium through 

r hich they conveyed their thoughts and deeds to their peers in 

iU ure was Greek, even though some of the records of these 

jghts and deeds have passed through Latin and Arabic 

,nslations before reaching us. In these records we find the 

;rms of theories and problems which have agitated the mathe- 

iatical world ever since, and of which some remain unsolved 

this day. We are told, indeed, that in mathematics most 

ads lead back to Hellas, and thus a book which makes any 

storical pretensions at all must needs begin with the question : 

r ho were these Ancient Greeks? 

The term conjures up in our minds a group of Aryan tribes, 
vvhich had originally settled on the southern part of the Balkan 
Peninsula and adjacent islands of the Aegean Sea; then, spread- 
ing out in all directions, eventually reached the shores of 
Asia Minor, Lower Italy and the African littoral. The insular 
character of the land and its maritime activities encouraged 
independence and local rule; yet, dwelling as these people did 
at the very gateway of Europe, and menaced as they continu- 
ally were by Oriental encroachment, they were often driven to 
"totalitarianism" as a means of self-preservation. Thus Ancient 
Greece became the proving ground of that struggle between 
oligarchy and democracy which has prevailed to this day; and 
as such we know it best. 



But this is just one aspect of the complex pattern which the 
term evokes in our minds. In Ancient Greece stood the cradle 
of our culture: literature and philosophy, architecture and 
sculpture, in the various forms in which these arts are cultivated 
today, all had their origin in Greece. The songs of her poets, 
the works of her sculptors and the tracts of her philosophers are 
not mere monuments of a glory that was, but sources of study 
and inspiration today and, probably, for many centuries to 
come. Nor was the genius of these people limited to the arts 
and letters; their penetrating insight into the mysteries of 
number, form and extension had led them to develop to a 
high degree of perfection a discipline which they named mathe- 
matics, and which was destined to become both the model and 
the foundation of all sciences called exact. 

This pattern becomes even more amazing when we con- 
template that all this magnificent culture was erected in a few 
short centuries. We are told, indeed, that this great intellectua 
upheaval had reached its peak in the fifth century B.C.; t" 
soon afterwards a general and rapid decline had set in, 
though a fatal blow had been struck at the very roots of t 
mighty tree, a blow from which it never recovered; that aftt 
lingering on for a few more centuries, vainly endeavouring t 
live up to the grandeur of its past, it had finally succumbed 
the coup de grace administered by rising Rome. 

Such is the picture of Ancient Greece as we perceive it through 
the thick historical fog of two thousand years. It is a perplexing 
picture, to say the least, for, as far as we know, nothing that 
ever happened before, or since, has even remotely resembled it. 
It is a picture of a people numerically insignificant, even when 
measured by standards of the Ancient World, which in the 
course of a few centuries erected a civilization of unprecedented 
magnitude, bequeathing to mankind for all time to come 
immortal treasures of literature, philosophy and mathematics. 
And the mystery becomes even more profound when we attempt 
as is indeed our duty to appraise this past in the light of the 



present. For the modern representatives of this ethnic group, 
far from exhibiting the acumen and finesse of their illustrious 
forebears, have contributed so little to the intellectual and 
artistic life of our time that it is difficult to conceive of any kin- 
ship between this Balkan people and the intellectual giants to 
whom our culture owes so much. 

Is there anything wrong with this picture? Could it be that 
it is but another cliche, one of the many synthetic products of 
the diversified industry which passes today for liberal education? 
Well, this much is certain: in so far as the history of mathematics 
is concerned, this conception of Ancient Greece calls for a 
wholesale and drastic revision. 

To begin with, the mathematical activity of Ancient Greece 
reached its peak during the glorious era of Euclid, Eratosthenes, 
Arfchimedes and Apollonius, a time when Greek letters, art and 
phjilosophy were already on the decline. There is a modern 
counterpart to this singular phenomenon. It was in the sixteenth 
ntury, the age of Cavalieri, Cardano, Galileo and Vieta, that 
mthematics was reborn; and the resurgence took place when 
he renaissance in arts and letters had already run its course, 
and the very names of Dante and da Vinci had become 
memories. Galileo was the central figure of that era, and the 
fact that he died one week before Newton's birth has been the 
subject of much historical comment. It is as significant, perhaps, 
that Galileo was born in 1564, within a few months of the 
death of the last great representative of Italian renaissance, 

In the second place, while Roman contributions to mathe- 
matics were less than negligible, there is no evidence what- 
soever that either the Republic or the early Empire had in any 
way hampered its progress. The eclipse of mathematics began 
with the Dark Ages, and the blackout did not end until the 
last Schoolman was shorn of the power to sway the mind of 


Lastly, it was not Greece proper but its outposts in Asia Minor, 
in Lower Italy, in Africa that had contributed most to the 
development of mathematics. Some of these outposts were 
Greek conquests, others had come under Greek domination 
through alliance or trade. Moreover, since the Greeks had no 
Gestapo to protect their Aryan blood from pollution, racial 
intermingling was widespread, and there is no evidence that 
these misalliances were frowned upon by Greeks of pure strain 

To be sure, the Greeks did divide mankind into barbarous 
and Hellenes; yet, scions of barbarian families who had adtpted 
Greek names and customs were viewed by Greeks as Helfnes. 
Thales of Miletus is a case in point. From all accounts h6vas 
of Phoenician origin, as was, indeed, Pythagoras; still, not oly 
was Thales classed by his contemporaries as a Greek >ut 
proudly hailed by them as among the seven wisest GreeksAs 
to his own attitude, listen to the words which one biograper 
puts in his mouth: "For these three blessings I am gratefuto 
Fortune : that I was born human and not a brute, a man id 
not a woman, a Greek and not a barbarian." 

In the centuries of Euclid and Archimedes Greek was tlu 
language of most educated men, whether they hailed from 
Athens, Syracuse, Alexandria or Perga. The significance of 
this will not escape the American observer familiar with the 
many melting pots strewn over this wide land. Could one assert, 
that a man was of Anglo-Saxon blood because he was named 
Archibald or Percival and enjoyed a good command of English? 
Is it not equally naive to contend that a man who lived in the 
third century B.C. was racially a Greek because he called himself 
Apollonius and wrote in Greek? 

The shores of the Mediterranean harboured many a melting 
pot into which Greeks and Etruscans, Phoenicians and 
Assyrians, Jews and Arabs were promiscuously cast. Who can 
tell today how the Aryans and Semites had been apportioned 
within these seething brews, or the Hamites, or the Ethiopians, 
for that matter? The motley mash passed into the sewers of 
history without leaving a trace of its composition behind it. The 



distilled essence alone remains, bottled in vessels which bear 
Greek inscriptions. 

"A dazzling light, a fearful storm, then unpenetrable darkness." 
So wrote Galois on the eve of his fatal duel; and if we did not 
know that he intended these words as a summary of his own 
short span of nineteen eventful years, we could take it as a 
description of the era of Hellenic mathematics. 

What brought about this brilliant progress, and what caused 
the subsequent eclipse? I shall not add my own speculations to 
those of Taine and Comte, and Spencer, and Spengler and 
countless other historians of culture. This much isjclear: mathe- 
matics flourished as long as freedom of thought prevailed; it 
decayed when creative joy gave way to blind faith and fanatical 

Chapter Two 


To Thales . . . the primary question was not What 
do we know, but How do we know it, what evidence can 
we adduce in support of an explanation offered. 


Six centuries before the zero hour of history struck, there thrived 
on an Aegean shore of Asia Minor, not far from what today 
exists as Smyrna, a group of Greek settlements which went under 
the collective name of Ionia. It consisted of a dozen or so towns 
on the mainland, of which Miletus was the most prosperous, 
and of about as many islands, of which Samos and Chios were 
the largest. When measured by present-day standards, the 
territory was so small that if modern Smyrna were run by 
American realtors all that was once Ionia would be reduced to 
mere suburban "additions" to the "greater city." 

Here, within the span of about fifty years, were born the twr 
"Founders" of mathematics, Thales of Miletus and Pythagoras 
of Samos. According to some of their biographers both were of 
Phoenician descent, which seems plausible enough since most 
of the coast of Asia Minor was at that time honeycombed with 
hoenician colonies. 

these Phoenicians? We remember them chiefly today 

^ntors of the phonetic script which was so vast an 

t over all the previous methods of recording experi- 

orinciple at least, it has undergone no significant 

wenty-five hundred years which followed. The 


Greek alpha, beta, gamma, as well as the Hebrew aleph, beth, 
gimel are but adaptations of the Phoenician symbols for these 

And yet, having bestowed upon mankind this marvellous 
method of recording events, the Phoenicians left practically 
no records of their own, and what little we know of them today 
we owe to Greek or Hebrew sources. They were a Semitic tribe, 
and their homeland was what we call today Syria. As Ganaanites, 
Moabites, Sidonians, they fill many a page of the Bible. 
Apparently, whenever they did not engage the Hebrews in 
mortal combat they fought them with subtle propaganda, 
inducing the fickle sons of Israel to abandon Jehovah for Baal 
and other more tangible gods. 

The Greeks knew the Phoenicians under a different guise. 
They spoke of them as crafty merchants and skilled navigators, 
and called them "Phoenixes," i.e., red, because of the ruddy 
complexions which the Mediterranean sun and winds had 
imparted to these ancient mariners. For, the Phoenicians roved 
that "landlocked ocean of yore" from end to end, exchanging 
wares and founding colonies, such as ill-fated Garthage, or 
Syracuse, the birthplace of Archimedes, who, reputedly, was 
also of Phoenician descent. 

i said tnat Thales was classed by the Greeks as one of the Seven 
Sages. Indeed, he was the only mathematician so honoured, 
and it was his reputed political sagacity and not his mathe- 
matical achievements that had earned him the title. Because of 
this distinction, Thales was the subject of many historical 
studies, with the result that much had been written on his life 
and deeds. Of what value are these biographical accounts? 
Here are some highlights from which you can draw your own 

We are told by one of these commentators that Thales was 
so keen an observer that nothing would escape his alert atten- 
tion; yet, according to another, he was so absent-minded that 
even as a grown-up man he had to be followed on his walks by 



his nurse lest he land in a ditch. We are informed by one that 
he was a seasoned merchant in salts and oils, and that it was the 
pursuit of this trade that had taken him to Egypt; but another 
tells us that he had come to Egypt as a very young man, and 
that, struck by the learning of the priests, he had tarried among 
them for more than a quarter of a century, returning to 
Miletus in advanced middle age. According to one account he 
had learned all he knew of geometry from these very priests ; 
according to another he was entrusted by the Pharaoh with 
the task of determining the height of the Great Pyramid, a 
problem which the priests had vainly tried to solve. 

The accounts on his views, social, political or philosophical, 
are just as confusing. Some tell us that he was a confirmed 
bachelor, that he had found an outlet for his paternal instincts 
in adopting his sister's family, that once when asked why he 
did not marry and have children of his own since he loved 
children so much, he replied: "Just because I love children so 
much." Other biographers, however, assure us that Thales had 
married and lived to be a patriarch, surrounded by children 
and grandchildren. Plutarch holds that Thales had democratic 
leanings, in support of which he quotes his letter to Solon. There 
Thales invites Solon to make his home in Miletus, apologizing, 
at the same time, that his native city is under the rule of a 
tyrant. Again, once when asked what was the strangest sight 
his eyes had perceived, he allegedly replied: "A tyrant ripe in 
years." But other sources have it that upon his return fron 
Egypt Thales made his home with the Milesian tyrant, tha 
for many years he acted as the latter's counsellor at large, am 
that it was, indeed, on the advice of Thales that the dictato 
had wisely declined a tempting alliance with Croesus. 

The accounts on Pythagoras may be less at variance, but they 
are bewildering enough in other respects, for, in addition to 
the confusing versions of the chroniclers, we have to contend 
here with disciples who would put into the mouth of their dead 



Master anything that fitted an occasion or proved a point. 
Indeed, Pythagoras became the centre of a cult which persisted 
for many centuries and exerted a tremendous influence on 
scientific and religious thinking. 

It is claimed that not only had he visited Egypt, but that his 
travels had taken him much farther East; that, in fact, much 
of the knowledge which he later conveyed to the Hellenes had 
been imparted to him by Persian Magi and the priests of 
Ghaldea. One is almost willing to believe this after examining 
the medley of views and taboos ascribed to Pythagoras. Yes, 
taboos since many of the rites of the sect later rationalized by 
his followers into principles have all the earmarks of taboos. 

A case in point is the alleged Pythagorean aversion to animal 
flesh. I say alleged, for, on this score, too, there is no unanimity, 
some biographers asserting that Pythagoras celebrated his 
mathematical discoveries by sacrificing oxen to the gods, while 
others go so far as to claim that he was the first to introduce 
meat into the diet of Greek athletes who hitherto had been 
training on figs and butter. 

Some followers of Pythagoras traced the interdiction to his 
doctrine of metempsychosis. According to them, the Master 
taught that of the three attributes of the soul only reason was 
exclusively human, while emotion and intelligence belonged to 
animals as well; that upon man's death his soul migrated from 
nimal to animal; and that, consequently, by killing an animal 
ne might mutilate a soul. All this makes beautiful reading but 
ails to explain why Pythagoras extended his dietetical pro- 
hibitions to beans. Indeed, according to Diogenes Laertius, this 
bean cult was the indirect cause of his death. Here is the story 
, for what it is worth. 

When Pythagoras returned from his Oriental travels, he 
found his native Sainos under the rule of a tyrant. He then pro- 
ceeded west and settled in Crotona, a prosperous city on the 
heel of the Italian Boot. There he eventually established a 
school and, incidentally, acquired great political power. Now, 
those were the days when totalitarianism was making serious 
inroads into Greek democracy, and so, as time went on, an 
opposition party arose which accused Pythagoras of dictatorial 
designs. A frenzied mob set fire to his mansion. The Master 



managed to escape, but having reached in the course of the 
ensuing pursuit a field of beans, he chose to die at the hands 
of his enemies rather than to trample down the sacred plants. 

The reader will have realized by now what a formidable task 
it would be to pick the few sound kernels from this biographical 
chaff, let alone use the material to analyse the achievements of 
the Founders. And yet, behind the hazy mist of these fanciful 
tales and legends the portraits of the two men emerge, mere 
silhouettes perhaps, but silhouettes that become much less elusive 
and confusing when viewed as parts of the larger panorama of 
classical mathematics. 

Neither Thales nor Pythagoras left any writings behind. In 
fact, the earliest mathematical work of any kind available to us 
today is Euclid's Elements. This does not mean that the mathe- 
maticians of the pre-Euclidean age had completely neglected 
to put their thoughts on parchment. On the contrary, The 
Mathematical Roster of Eudemus mentions two textbooks on 
mathematics written within one hundred years of Thales' 
death ; one of Anaximander of Miletus, a pupil of Thales, the 
other by Hippocrates of Chios. However, these works, as man* n 
others of that period, were lost in the course of the next two 
thousand years. The same fate was shared by at least two works 
of Euclid, and by many tracts and treatises of the post- 
Euclidean period. 

Now, most of these lost writings were still available in the 
fourth century A.D., when Pappus of Alexandria wrote his 
great book The Synagogs (Collection). What happened to this 
sizable literature? A few of these found their way to Arabia and 
were eventually restored to Europe, thanks to the enlightened 
care of Moslem scholars ; others were destroyed when the great 
Alexandrian library was burned to the ground on the order of 
Bishop Theophilus, Anno Domini 392; still others perished in 
the darker ages which followed. 

As a result, the only sources which could throw light on the 



pre-Euclidean period are some passages from the Dialogues of 
Plato; some casual remarks found in the writings of Archimedes, 
Apollonius, Hero and others ; the Collection of Pappus which 
may be viewed as a sort of encyclopaedia of Greek mathe- 
matics; and fragments of an already mentioned essay on the 
history of mathematics written by a contemporary of Euclid, 
one Eudemus of Rhodes, as quoted by the Platonist com- 
mentator Proclus seven centuries later. 

Unfortunately, Plato's opinions on matters mathematical were 
vitiated by a number of factors. In the first place, he was a 
Pythagorean par excellence. Try as he might, he could not free 
himself of the tendency to attribute most achievements in philo- 
sophy and in mathematics to either Pythagoras or his adherents. 
What is more, there is little evidence that he tried to free himself 
of that bias. Thus, we find Thales barely mentioned in the 
Dialogues; as to Hippias, Hippocrates and Democritus, who had 
either kept aloof from the School or had openly opposed it, 
they were treated with contemptuous silence. 

In the second place, despite the vociferous claims of the 
^latonists and Neoplatonists, Plato was not a mathematician, 
^o Plato and his followers mathematics was largely a means to 
in end, the end being philosophy; they viewed the technical 
Aspects of mathematics as a mere device for sharpening one's 
wits, or at most as a course of training preparatory to handling 
the larger issues of philosophy. This is reflected in the very 
name "mathematics," a literal translation of which is a course 
of studies or, as we would say today, a curriculum. It was in this 
sense that the term was used in the Platonic Academy, and it 
was not until later that mathematics became the name of the 
science of number, form and extension. 

What interested Plato most in mathematics were the meta- 
physical issues which lay back of its concepts. The very topics 
treated in the Dialogues give eloquent evidence to that effect. 
Thus, the so-called Pythagorean proposition which binds the sides 



of a right triangle received but casual treatment in the Dialogues, 
and even then the emphasis was on the number-theoretical 
aspects of the problem. On the other hand, such topics as 
harmony, triangular numbers, figurate numbers, and many other 
themes which we view today as more or less irrelevant, if not 
trivial, were taken up at length. Indeed, the guiding motive 
behind the mathematical predilections of the Pythagoreans 
and Platonists was of a type which the professional philosophers 
call metaphysical but which for the nonprofessional have all the 
earmarks of the occult. 

The Dialogues were written more than one hundred years after 
the death of Pythagoras. In the course of these one hundred 
odd years the doctrines propounded by the Master had been 
vigorously attacked, first by the Ionian philosophers who fol- 
lowed in the footsteps of Thales, then by the Sophists who were 
led by Parmenides of Elea. The critique of the opponents and 
the defence of the proponents fill many an eloquent page of the 
Dialogues, and these, vivid pages paint a far more convincing 
portrait of Pythagoras than do the extravagant tributes of his 
zealous followers. 

These pages reveal a religious mystic who viewed number 
as the key to the plan which the Supreme Architect had usec 
in fashioning the universe. Be it the movement of heavenly 
bodies or the composition of matter, the structure of thought or 
the principles of human conduct, everything was expressible 
in number because all was governed by number. It was the 
philosopher's mission to interpret the work of the Creator by 
deciphering, as it were, the intricate scroll of creation; but to 
do this, he must needs first master the code in which this scroll 
was written, and this code was mathematics. 

"Number rules the universe." Did Pythagoras foreshadow 
in this dictum the vast system of formulae and equations by 
means of which modern science links the phenomena of nature, 
as Galileo foreshadowed it two thousand years later, when he 
wrote: "The universe is the grand book of philosophy. The 



book lies continually open to man's gaze; yet, none can hope 
to comprehend it who has not first mastered the language 
and the characters in which it is written. This language is 
mathematics; these characters are triangles, circles and other 
geometrical figures." 

Or did Pythagoras conceive the scheme of creation as a sort 
of supernumerology which assigned to everything material or 
spiritual an integer, and reduced the relations between things 
to operations on these integers? Most of us today would view 
these two attitudes as irreconcilable, extolling the first as 
scientific and branding the second as occult. But to Pythagoras, 
and to his followers for centuries to come, the line of demarca- 
tion was by no means so sharp. 


We find among the writings of the Pythagoreans such 
undisguised numerology as this : two being the number of man 
and three that of woman, Jive must of necessity be the number of 
marriage) since marriage is the union of the two sexes ; or that 
perfect numbers are symbols of perfection, human or divine, 
since such a number is the sum of its divisors , hence self-contained 
ind complete, i.e,, perfect. In music, astronomy and geometry 
his numerology assumed more subtle forms. For instance, 
instead of representing a geometrical figure by a single number, 
the Pythagoreans assigned to it a number type. This accounts 
for their extraordinary interest in triangular, square and figurate 
numbers which appear so utterly inconsequential today. 

But we also discover in the Pythagorean speculations more 
than a mere germ of what, for want of a better name, we call 
today the scientific attitude. That this attitude remained in an 
embryonic stage was due to the state of algebra at that time. 
Today, the representation of a physical law by means of a 
formula is so common that we accept it as though it were 
granted to man by Providence. But far from it being a gift 
from heaven, it was the culmination of a long and painful 
evolution, since even the most simple formula implies ideas 



which in the days of Pythagoras were either in their infancy or 
did not exist at all. 

Indeed, in the first place, any formula postulates the existence 
of the rational, if not of the real, domain of numbers, while in the 
days of Pythagoras number meant positive integer and nothing else. 
In the second place, the reason we view the formula as the 
ultimate goal in the study of a phenomenon is because the 
problem is then reduced to the routine operations of arithmetic and 
algebra which most of us have learned on the school bench. 
Remember, however, that when about the year 1600 Vieta 
introduced literal notation, it struck even mathematicians as a 
sensational innovation; also, that even the operations of ele- 
mentary arithmetic are barely more than five hundred years 
old ; then transfer yourself in imagination to the sixth century 
B.C. when rhetorical algebra was in its very infancy, and positional 
numeration not even a dream. Again, before a formula linking 
the various entities can even be as much as formulated, it is 
necessary that these entities be measurable, or at least conceived 
as numbers; but in the days of Pythagoras, and for many 
centuries to come, such entities were hardly more than figures 
of speech or qualitative attributes. 

Geometry was the one field where the transition from the 
qualitative to the quantitative stage was well advanced, and so it 
is not surprising that geometry became the proving ground for 
the Pythagorean number philosophy. 

This brings us to the so-called Pythagorean theorem the relation 
between the sides of a right triangle which we express succinctly 
today in the formula, 

I shall deal with the mathematical facets of this important 
proposition in subsequent chapters. Here, we are concerned 
with the historical aspects of this relation. How much did 



Pythagoras contribute to the proposition which bears his name? 
Was he the discoverer of this property of right triangles? Was 
he the first to point out its far-reaching implications? Was he 
the first to demonstrate the theorem by logical arguments 
applied to the basic axioms of geometry? Well, such historical 
evidence as is available to us today suggests that all these ques- 
tions be answered in the negative. 

Pythagoras could not have been the discoverer of the relation, 
because, in one guise or another, this property of right triangles 
was known and used by scholars and artisans of Oriental lands 
thousands of years before Pythagoras was born. Indeed, we 
must bear in mind that while deductive geometry is barely more 
than twenty-five hundred years old, empirical geometry is 
probably as old as civilization itself. Many geometrical rela- 
tions which were eventually confirmed by deductive reasoning 
had been known as experimental facts thousands of years before 
the Greeks began to cultivate geometry as a science. This is 
attested by much documentary evidence, such as the clay tablets 
of the Babylonians and the Egyptian papyri. But even more con- 
vincing is the mute testimony of the Pyramids and of the ruins 
of ancient edifices uncovered by archaeological research. Indeed, 
it is not conceivable that such structures could have been 
designed or erected without a considerable knowledge of prac- 
tical geometry. 

Again, if by proof of a mathematical proposition we mean 
establishing its logical validity on the basis of a set of assump- 
tions accepted as self-evident, then Pythagoras did not possess 
a proof of the theorem which bears his name; not because such 
a proof was beyond the ken of his period, but because he was 
temperamentally uninterested in proofs of this nature, as may 
readily be gleaned from the methods which he used in his 
numerological deductions. 


I am about to venture a conjecture, and I want to take the 
opportunity to emphasize that by advancing this and sundry 
other opinions which will appear in this work / neither invoke 



authority nor claim originality. I do not invoke authority, because 
in the absence of authentic documents opinion rests on imagina- 
tion, and one imagination is as good or as bad as another. I do 
not claim originality, because to be valid such a claim would 
imply an exhaustive knowledge of the literature of the subject, 
and this would smack of omniscience. My position in matters 
of history is that one must either be prepared to substantiate 
a statement by reference to written records or else honestly 
admit that it is one's own conjecture and shoulder the 

With regard to the Pythagorean theorem my conjecture is 
that at least in one of its several forms the proposition was known 
before Pythagoras and that and this is the point on which I 
depart from majority opinion it was known to Thales. I base 
this conjecture on the fact that the hypotenuse theorem is a 
direct consequence of the principle of similitude, and that, 
according to the almost unanimous testimony of Greek his- 
torians, Thales was fully conversant with the theory of similar 

I do not contend that Thales was aware of the vast implica- 
tions of this proposition for geometry and mathematics generally. 
These implications were not and could not have been fully 
appreciated until the advent of analytic geometry. But since, 
by the same token, Pythagoras could not have been aware of 
the geometrical significance of this theorem, the question as to 
whether he was the first to recognize the mathematical impli- 
cations of the proposition which bears his name should also b 
answered in the negative. 

On the other hand, there is no doubt that Pythagoras full 
appreciated the metaphysical implications of this relation. For, 
the fact that the sides of a right triangle were connected by a 
law expressible in numbers suggested that all geometrical 
entities responded to such numerical laws, since, in the last 
analysis, any such entity could be viewed as an element of a 
rectilinear figure which, in turn, could be resolved into triangles, 
and any triangle was made up of right triangles. Thus this 
relation, which Euclid regarded as a mathematical theorem and 
which we construe today either as a postulate of geometry or 
as a ready consequence of other such postulates, was to 



Pythagoras and the Pythagoreans a basic law of nature, and, 
at the same time, a brilliant confirmation of their number 


With the possible exception of Aristotle, no other philosopher 
of antiquity received as much publicity as Pythagoras. The 
spectacular character of the man, the fact that he was the 
titular head of a semi-religious cult and the acknowledged 
fountainhead of the Platonist School may explain his wide- 
spread fame. I spoke before of the extravagant claims made for 
him by his followers. These claims were not confined to the 
realm of mathematics. Among other things, he was credited 
with being the originator of the heliocentric hypothesis, in spite of 
the undisputable evidence that this hypothesis had been first 
propounded by Aristarchus, a contemporary of Archimedes. 

The belief that Pythagoras had taught that the earth revolved 

about the sun persisted even after the contributions of Coper- 

"cus had been made public. In fact, in 1633, when Galileo 

s tried for his heresies, the immortal document of the 

}uisition, in listing his errors, called the heliocentric hypothesis 

'ythagorean doctrine. This brilliant subterfuge spared the 

ly Office the embarrassment of indicting Copernicus, an 

.ained priest of the Church. 

Today no one would think of associating Pythagoras with 

^ Copernican theory; yet, although his discovery of the 

ation in a right triangle had no stronger foundation in fact, 

2 idea that he was the originator of the theorem stubbornly 

;rsists. The result is that the name of Pythagoras is a house- 

>ld word to most educated people, while the name of Thales 

known to but a few specialists, and even these regard him as 

. philosopher rather than a mathematician. 

The Pythagorean theorem is by no means the only case when 
.he honour for a capital achievement has been conferred on 
he wrong man, nor is mathematics the only field where such 
'miscarriages of justice" have occurred. The study of this 
.phenomenon is a fascinating chapter in the history of thought 


but lies beyond the scope of this book. However, as we proceed, 
we shall encounter other episodes of the kind which may cast 
some light on the underlying causes. 

I would not advocate that the Pythagorean theorem be 
renamed Thalesian, even if I had sufficient documentary evi- 
dence to support my conjecture. The term " Pythagorean" has 
by now become a part of mathematical nomenclature. That 
this nomenclature is far from perfect, that, in fact, it bristles 
with misnomers, is generally recognized. Still, nobody seems 
to be able to do anything about it not that reforms have not 
been proposed, but that almost invariably it was found that 
the effort required to effect the change by far outweighed 
the advantages gained. 


The more one attempts to appraise the mathematical achieve- 
ments of Pythagoras, the less impressive they appear. On the 
contrary, the more one studies the period of Thales the more 
one compares the knowledge he bequeathed to posterity wi* 1 
the one he had found when he began his work the mo 
does his mathematical stature grow, until one is impelled 
range Thales with such figures as Archimedes, Fermat, Newtc 
Gauss and Poincare. 

But while we cannot place Pythagoras among the great 
even near-great mathematicians, his position in the history 
scientific thought remains unchallenged. To be sure the dictu 
"Number rules the universe" might bring a condescendii 
smile to the lips of a modern scientist : yet, forget the lofty for 
in which these words were put; conceive numbers not as ju, 
positive integers, as the Pythagoreans did, but in the broa 
modern sense of the term : then is there anything in the dictun 
to which a modern scientist could not or would not subscribe: 
The theories of relativity and quanta have shaken the physical 
sciences to their very foundation, forcing the physicist to cast 
overboard such principles as conservation of energy or economy oj 
action, and to revise the very concepts of space, time, matter 
cause and effect. Still, number reigns as firmly in the new physics as 



it did in the old. The argument that the study of any phenomenon 
has not been consummated until the phenomenon has been made mathe- 
matically articulate is as convincing today as it was in the days 
of Pythagoras whilst the conjecture that physical attributes may 
exist that are beyond the power of number to express would be 
as odious to the man of science today as it was to Pythagoras. 


Chapter Three 


Geometry in every proposition speaks a language 
which experience never dares to utter and of which, 
indeed, it but half comprehends the meaning. 


Let us first of all dispose of some chronological matters pertain- 
ing to the era which, for want of a better name, we call classical 
mathematics. Sharing the average reader's keen dislike for 
dates, I shall confine myself to the "century posts" of that long 
and tortuous trail. 

The year 600 B.C. finds young Thales in Egypt eagerly 
absorbing the knowledge of its priests : what appears to us as 9, 
maze of rules, recipes and rites amassed without rhymef or 
reason looms to the young Greek as a token of inestimable 
promise; 500 B.C. Pythagoras is at the height of his glory at 
his school in Crotona; 400 B.C. young Plato is fleeing Athens 
lest the fate of his teacher, Socrates, overtake him; 300 B.,Q. 
Euclid's Elements usher in the great century of classical ma ,the- 
matics, the century of Archimedes and Eratosthenes; 200 
B.C. Apollonius completes his monumental treatise on & onic 
sections; 100 B.C. here, roughly, may be placed the birth of 
Hero of Alexandria; A.D. 100 here, roughly, may be pla< :ed 
the birth of Ptolemy; A.D. 200 darkness is already settl \ng 
over the Hellenic world; Diophantus, then Pappus; last flicke rq 
of a dying fire, a fire which is not to be rekindled for anothe 
thousand years. 

We are concerned here only with the first three centuries oi 
this near-millennium; yet, this relatively short period is the 



most perplexing in the history of mathematics. Not that it is 
so crowded with achievement or bristles with great names, as 
does the century that follows Euclid. No, it is not wealth of 
material but scarcity of information that forces a would-be 
interpreter of that period to resort to conjecture and specula- 
tion. The wisest course would be to dismiss the whole matter 
with a few summary remarks. Unfortunately, one cannot thus 
dismiss the fact that it was during these centuries that geometry 
had come of age ; and so the story must be told, even if at the 
risk of some speculation and conjecture, and I know of no better 
way of telling it than by reversing the chronological order of 
events, proceeding in retrogress, as it were. Accordingly, I 
begin with Euclid. 

He lived in Alexandria. These four words sum up all we know 
of the life of the man who was instrumental in shaping the 
mathematical education of countless generations, and one of 
whose works has had, next to the Bible, perhaps the largest 
circulation of any book ever written. No legends trail his name, 
and even the anecdoters have spared him. He was a prolific 
writer, yet only one of his works withstood the wear and tear 
of the centuries that followed him. The Greeks rarely referred 
to him by name: to them he was simply the author of the 
Sro^e^, the work we call Elements, although Basic Principles 
would have been a more fitting translation of the Greek title. 

The book was primarily a treatise on geometry, even 
though it did deal with other topics, such as perfect numbers, 
primes and irrationals. As a treatise on geometry it was so 
thorough that it serves to this day as the basis of most of our 
elementary textbooks. Still, comprehensive though it was, the 
Elements apparently did not express fully what Euclid knew on 
the subject, since at least two of his lost books also dealt with 
geometry. One of these, Conies, must have been an exhaustive 
study indeed, because when about a century later Apollonius 
came out with his own treatise on these curves, some of his 
more determined antagonists accused him of having plagiarized 
Euclid's work. 



The title of the other lost book was Porisms, and all we 
really know about its contents comes from references to it by 
Pappus. This passage inspired Fermat to undertake a restora- 
tion of the lost work; unfortunately, the version of this great 
master of the seventeenth century suffered the fate of the origi- 
nal, as did several other attempted restorations, notably those of 
Wallis and of Simson. 

Thus the only thing certain about the Book of Porisms is its 
title, and even here there is no general agreement as to the 
sense in which Euclid used the term. The literal translation 
of TTOpicrjuocr is method, means, implement, and it is quite con- 
ceivable that Euclid meant just that, i.e., that the Book of 
Porisms was intended as a sort of supplement to the Book of 
Elements, that it implemented the fundamental principles 
expounded in the treatise by practical methods of construc- 
tion. This would not contradict the assertion of Pappus that 
the book dealt with loci, inasmuch as the locus was the basic 
device of all Greek construction. 

On the other hand, it is just as possible that the Book of 
Porisms was Euclid's contribution to those celebrated problems of 
antiquity which agitated the minds of all mathematically inclined 
individuals of his day. These problems were : squaring the circle, 
doubling the cube, trisecting any given angle and cydotomy* i.e., 
dividing a circle in any given number of equal parts, the construction 
in every case to be executed by the exclusive means of the straight- 
edge and the compass, or what amounts to the same thing 
by introducing no other auxiliary loci than straight lines and 
circles. The fact that none of these problems was mentioned 
by Euclid in the Elements would lead one to surmise that he 
had intended to deal with them elsewhere. 

All these problems have been solved in modern times, i.e., 
solved in the "negative" sense, by demonstrating that the 
constructions cannot be executed within the restrictions 
imposed. The proofs require resources of algebra and analysis 
which the Greek mathematicians did not possess ; this, however, 
could not have prevented a Euclid from surmising the truth, 



or from endeavouring to prove it by such methods as were at 
his disposal. Now, in the absence of an articulate algebra, the 
most natural approach to the question as to which construc- 
tions could be executed by ruler and compass and which could 
not was through a systematic study of geometrical procedures 
and of the loci which they generate, and, for all we know, it 
was just such a study that the author of the Book of Porisms 
had in mind. 

This much is certain: Euclid had at his disposal a vast store 
of mathematical knowledge, particularly in the field of geo- 
metry. How much of this was his own discovery, how much 
the work of his contemporaries, or the bequest of an earlier 
age? There is irrefutable evidence that a substantial portion 
of the material recorded in the Elements was known before 
Euclid, and there is nothing either in the style or in the plan 
of the treatise to suggest that it was intended as a collection 
of original contributions. Thus, on the whole, it is safe to 
assume that the chief objective of the author of the Elements 
was to put system and rigour into the work of his predecessors. 

Who were these predecessors? Well, the roster of the fourth 
century contains such names as Archytas, Eudoxus, Menaech- 
mus, talented, even brilliant, men who undoubtedly had 
exerted considerable influence on the mathematics of their 
own period. It is unthinkable that their work had failed to 
influence Euclid as well, more particularly in his treatment 
of solid geometry. And yet, it would be equally erroneous to 
attribute to these men the discovery of the basic propositions 
of plane geometry which, after all, constitute the very lifeblood of 
Euclid's Elements, because all indications are that they had 
themselves inherited the rudiments of plane geometry from an 
earlier age. Indeed, unless they had these rudiments at their 
very finger-tips they could not have made the discoveries with 
which they were credited by subsequent commentators. 

Let us tarry for a while on these achievements. Archytas is 
supposed to have been the first to study geometry on a circular 
cylinder, discovering in the process some of the properties of 



its oblique section, the ellipse. Eudoxus was the first to study 
geometry on a torus, i.e., the surface generated by the circum- 
ference of a circle which revolves about an axis in its plane. 
He discovered the sections of this surface by planes parallel to 
the axis of revolution, quartic curves which today are called 
Cassinian ovals, after the French-Italian astronomer, Giovanni 
Cassini, who had advanced the theory that Kepler's ellipses 
were mere approximations, while the true planetary orbits were 
these very ovals. Finally, Menaechmus 5 claim to fame was the 
discovery of the conies, i.e., the plane sections of a circular cone. 

Observe that all these studies involved three-dimensional 
considerations, despite that in each case the geometer had 
the properties of a plane locus as his avowed objective. It is 
as though he felt that the basic material of plane geometry 
had been exhausted, and that further progress could be 
achieved only by envisaging a plane locus as one embedded 
in space. But why did these geometers single out these special 
loci for their considerations? The answer takes us back to the 
celebrated problems which I mentioned in the preceding 
section, more particularly to that of doubling the cube which the 
ancients called the Delian problem. 

The Delian problem was described with eloquence and humour 
in a letter which Eratosthenes addressed to King Ptolemy. 
The purpose of the letter was to present his own solution of 
the problem, a device which he called mesolabe. The following 
is a free translation of the opening paragraph of the letter. 
The complaint of Glaucos quoted by Eratosthenes comes 
from Euripides' lost tragedy Poleidos. 

"An ancient playwright, describing the tomb of one hundred 
square feet which Minos had erected for Glaucos, put into the 
mouth of the latter the words, 'Too small hast thou built my 
royal tomb: double it but abide by the cube. 9 Geometers long sought 
to determine how a given body may be doubled without 
altering its form, and called this problem Doubling the Cube. 



There was much confusion for a while; at long last Hippocrates 
of Chios showed that the problem could be solved if one but 
knew how to insert two mean proportionals between a recti- 
linear segment and its double, whereupon one perplexity gave 
way to another no less perplexing. 

"Then, the Delians, afflicted with a scourge, consulted an 
oracle who ordered that the altar in one of their temples be 
doubled. In their perplexity the Delians appealed to the 
geometers trained in the Platonian Academy, who zealously 
proceeded to solve the problem by seeking to insert two mean 
proportionals between two given segments. Archytas of Taren- 
tum achieved this by means of a cylinder, and Eudoxus of 
Cnidus by means of the so-called oval curves. But while their 
methods lack nothing in geometrical rigour, their designs are 
not readily amenable to construction by hand . . . The design 
of Menaechmus is more handy, but it, too, is quite laborious." 

Our backward march takes us next to the fifth century. Is this 
to be the end of our quest? Were the principles on which 
Euclid later erected his Elements discovered by the mathe- 
maticians of that period? No, the geometers of the fifth century 
seem to have been even less preoccupied with the rudiments 
than were those of the fourth. They, too, took the Elements 
for granted; they, too, were irresistibly drawn by the mirage 
of the unsolved problems. Indeed, if the fourth is to be called 
the Delian century of mathematics, the fifth was the century 
of the circle-squarer. 

The lure of the quadrature problem was not confined to 
professional mathematicians like Hippias or Hippocrates; 
nor even to near-mathematicians, such as the philosopher 
Anaxagoras, teacher of Pericles, who, according to legend, had 
whiled away his time in prison by working on the problem. 
It attracted scores of amateurs and notoriety seekers. From all 
evidence, it was the fifth century that witnessed the emergence 
of that strange species whom Augustus De Morgan nicknamed 
pseudomath. This barnacle has clung to the hull of mathematics 



throughout its long and eventful voyage. It persists to this 

Today, we identify the problem of squaring the circle with 
determining the mathematical character of the number n, on the 
ground that the area of the square sought is equal to nR 2 . 
This reduces the quadrature problem to the construction of 
the segment nR, the segment R being given. Is the number 
rational, i.e., is n a solution of some linear equation with integral 
coefficients? If not, is it a solution of some quadratic equation, 
or of a chain of such equations? If not "quadratic," is it at 
least algebraic, i.e., is it a root of some irreducible equation of 
higher degree? Any number which is not a solution of an algebraic 
equation with integral coefficients is said to be transcendental. 
How difficult such questions might be, can be judged from the 
fact that it took nearly twenty-four hundred years to establish 
the transcendental character of the number n. 

But what has the character of a number to do with the 
construction of a figure? Reserving a more satisfactory answer 
for a later chapter, let me say here that it may be shown that 
if a number n is rational or quadratic, then the segment nR may be 
derived from the segment R by ruler-compass operations. More 
generally, if n is an algebraic number, then it is possible to 
construct the segment nR by means of a more or less com- 
plicated linkage, i.e., by some device made up of rigid bars and 
pivots. However, no linkage exists which would permit one to 
derive nR from R, if n is a transcendental number; the mechanism 
would have to contain, in addition to bars and pivots, such 
members as rollers, cams, gears, etc. 

When applied to the celebrated problems, these considera- 
tions permit us to conclude that since the number rr is transcen- 
dental, the quadrature problem cannot be solved by straightedge and 
compass, nor by any sort of linkwork, for that matter; on the other 
hand, such problems as the duplication of the cube, the multisection 
of a general angle, the division of a circle into any number of equal 
parts are amenable to algebraic equations and can, therefore, 
be solved by means of linkages. 



The major cause of the prevailing confusion in regard to such 
questions as the trisection of an angle or the squaring of a 
circle is failure to discriminate between the problem of determining 
the character of a number and that of evaluating the number. This 
confusion is by no means limited to laymen. Thus, writers who 
certainly should know better have asserted that the origins 
of the quadrature problem may be traced to China, Babylon 
or Egypt, when all they mean is that architects and surveyors 
of those ancient lands, confronted with the necessity of measur- 
ing circular arcs and areas, were led to assign some rational 
value to the number TT. 

Now, granted that attempts to evaluate the ratio of the 
circumference of a circle to its diameter are as old as empirical 
geometry, and that the latter is, probably, as old as civilization 
itself, such attempts have nothing to do with the problem of 
determining the mathematical character of TT or with its geo- 
metrical counterpart, the squaring of the circle problems 
which acquired meaning only after geometry had emerged 
from the empirical into the deductive stage. Even at that, 
there is doubt that such questions were actually raised during 
the earlier period of deductive geometry, since a precise 
formulation of these problems implies a thorough knowledge 
of the fundamental propositions of plane geometry, a mastery 
of its basic constructions and, above all, a critical attitude 
which comes only with advanced geometrical rigour. 

We may be reasonably sure that the quadrature problem 
was born on Greek soil, and while we shall probably never 
know who first proposed it, it is certain that the event took 
place not later than the middle of the fifth century. Indeed, 
we know that the problem had inspired the efforts of the two 
mathematicians who dominate the second half of the fifth 
century, namely, Hippias of Elis and Hippocrates of Chios, 
the same Hippocrates who was mentioned by Eratosthenes in 
connection with the Delian problem. It was Hippocrates who 
first brought out that squaring the circle and rectifying its 
circumference were two horns of the same dilemma. Hippias 



went even further by devising a genuine "mechanical" quadra- 
ture of the circle, the only one of its kind in classical times, and 
one which for ingenuity, insight and rigour would do honour 
to an Archimedes.* 


These are the facts: Hippias was born about 425 B.C.; Thales 
died about 550, after an active career which extended over 
almost half a century; it is reasonably certain that deductive 
geometry as such did not exist either in Greece or elsewhere 
prior to the birth of the Founder. Thus in less than two hundred 
years geometry had undergone a complete metamorphosis, 
changing from an Egyptian hodgepodge of rules of thumb 
to a full-fledged discipline. 

This extraordinary progress becomes even more astounding 
when we reflect that, in so far as propagation of ideas is con- 
cerned, those two centuries were like two decades in our 
own time. Remember, indeed, that the habit of putting down 
one's thoughts in writing was practically nonexistent in those 
days; that such manuscripts as did live to see the light of day 
could be reproduced only by laborious copying; that mathe- 
matical nomenclature was in its infancy, and symbolism did 
not exist at all, since even the designation of vertices of figures 
by letters of the alphabet was not known before Hippocrates; 
that in the absence of a centre of mathematical activity such 
as Alexandria in the post-Euclidean centuries, mathematics 
was being cultivated in widely separated regions; that the 
exchange of ideas among scholars was largely by personal 
contact, and that a journey from Asia Minor to Lower Italy, 
which today can be accomplished in less than three hours by 
plane, required then many a month. 

And this is not all. When these two centuries are viewed in 
the light of actual achievement, they dwindle into at most 
one, for the span of one hundred and twenty-five years which 
separates Thales from Hippias was particularly barren of 
mathematical progress. Indeed, it produced only one mathe- 
matician of note, namely Pythagoras, and he and his disciples 

* See Chapters 10 and n. 


had been too preoccupied with the occult and metaphysical 
aspects of mathematics to contribute much of value to geo- 
metrical technique. 

Thus the conclusion is fairly forced upon us that this pro- 
digious achievement was the work of one man: Thales of 
Miletus. "He endowed geometry with rigour, and founded it 
on congruence and similitude" such is the generous testimony 
of one historian. But this appraisal is not generous enough, 
for, he also implemented these principles with a rich tech- 
nique, and taught how to apply this technique to construction 
and proof. That it took the Greeks more than a century to 
absorb the work of the Founder becomes less surprising when 
we contemplate the grandeur and revolutionary character of 
his achievement and remember that there were no geometers 
when Thales began: Thales the teacher produced the first 
geometers, even as Thales the thinker founded the first geometry 
worthy of the name. 

The alternative theory offered in explanation of the rapid 
growth of Greek geometry is that much of the achievement 
claimed for the Greeks was actually of foreign origin. This 
theory is relatively new; as a matter of fact, any such conten- 
tion would have found little support among the historians of 
science of the nineteenth century. To be sure, classical com- 
mentators on mathematics had been rather vociferous in 
acknowledging the debt which Greek geometry owed to the 
priests of Isis and Osiris. However, their effusive appraisal of 
the Egyptian contribution has not been borne out by the 
papyri deciphered in the course of the last century. Indeed, 
these documents reveal that Egyptian geometry, even when 
considered as an empirical effort, was so rudimentary, if not 
crude, that the Greeks could not have conceivably derived 
anything worth while from that source. 

However, considerable progress has been recently made in 
deciphering the cuneiform inscriptions on the clay tablets 
discovered among the ruins of ancient Babylon. These studies 



have disclosed that the Babylonians possessed a much greater 
store of scientific knowledge than has hitherto been suspected. 
In arithmetic^ for instance, they had devised a positional numera- 
tion and even a symbol equivalent to our decimal point; 
in algebra^ they knew how to set up quadratic and cubic equa- 
tions, and had even contrived to solve such equations by means 
of elaborate numerical tables ; finally, in geometry, too, they had 
by far excelled the Egyptians in measuring areas and volumes. 
These archaeological studies have led some writers to conjecture 
that Babylonian learning had in some way infiltrated into 
Greece, say, during the sixth century or even before, which 
could account for the extraordinary progress of Greek mathe- 
matics in the ensuing centuries. 


It is not my purpose here to examine this hypothesis in detail. 
After all, it matters little where the Greeks obtained the geo- 
metrical material on which they eventually erected their 
geometry. The issue is where, when and how the method of 
deductive reasoning emerged to turn geometry into a mathe- 
matical discipline. The ' 'Babel" theory would acquire signi- 
ficance only if its proponents could establish that the Babyl- 
onians had arrived at their geometrical rules by means of 
deduction, and thus far the cuneiform inscriptions of the 
Babylonians have yielded no greater evidence to that effect 
than the hieroglyphics of the Egyptians. 

There are several other questions which the supporters of 
the Babel hypothesis will have to answer before they can 
establish their theory on a sound footing. How is it that we 
find no mention of Babylonian influence in any of the accounts 
given by Greek commentators? If this was a deliberate and 
concerted attempt to conceal the sources of Greek erudition, 
why were the same men so generous in acknowledging their 
debt to the Egyptians? If the influence existed, why did it 
leave no trace in the field where the Chaldeans were most 
proficient and the Greeks most deficient, namely, in algebra? 
Certainly, in an age when the problems of doubling the cube 



and trisection were so much in vogue, the Babylonian methods 
of handling cubic equations could have been used with success. 


I set out to trace the evolution of Greek geometry from its 
inception to the days when it had grown to full stature in the 
Elements of Euclid. This survey has revealed that the dominating 
motif of the first and last phases of that era was geometry as a 
discipline, while the stimulus of the intervening period was the 
challenge of the unsolved problems. The history of that era is 
like a symphony the finale of which is but a variation on the 
introductory theme, while the intermediate movements are 
built on quite a different motif, with the original theme as a 
mere accompaniment. 

In a certain sense this survey was the defence of a thesis, and 
I fear that in spite of the arguments adduced in its support, 
the thesis will strike most of my readers as indefensible. Indeed, 
here is a comprehensive and well-integrated discipline which, 
having survived without appreciable change for more than 
two thousand years, remains today one of the bases of universal 
education. The idea that this achievement was the work of a 
single man, or even of a single generation, is so much at 
variance with accepted ideas on the progress of knowledge 
that it seems to border on the fantastic. 

It does border on the fantastic, and in all fairness I must 
warn the reader that such fantasies await him at every twist 
and turn of mathematical history. I once attended a lecture of 
Poincare during which he made the off-record remark that 
the history of mathematics resembles an anthology of amazing 
stories, and that geometry was the most amazing of the lot. 
So, I shall not attempt to strengthen my thesis with further 
argument, trusting that the reader who perseveres with me 
on these exotic excursions into the shadowland of number, 
form and extension will become immune to shocks and will 
finish by realizing that his original reaction has largely been 
due to preconceived notions on the progress of knowledge. 

With this I rest my case. 


Chapter Four 


It is not that history repeats itself, but that historians 
repeat each other. 


We journey to the valley of Giza, one-time burial ground of the 
ancient Pharaohs. The huge tombs of these long-forgotten 
potentates still stand, enduring monuments to their colossal 
vainglory and to the technical prowess of an age which ante- 
dates Greek civilization by more than four thousand years. 
Among these structures, sprawling over more than twenty 
acres and rising to nearly five hundred feet above its base, 
looms the Pyramid of Cheops, surnamed the Great. 

In this grandiose setting was taught, according to Greek 
legend, the first lesson in deductive geometry. The time : about 
600 B.C. The pupils: the venerable priests of Isis and Osiris, a 
cult so old as to appear rooted in eternity. The teacher: one 
Thales, a Greek who had come to those shores with the express 
purpose of wresting from these very priests the secrets of their 
mystic knowledge. The problem: to determine the altitude of the 
Great Pyramid. 

Why did the priests put the problem to Thales? Was it a 
challenge designed to put the upstart Greek in his place? Were 
they unable to solve the problem by their own efforts? Is it 
conceivable that a people who had displayed such skill in 
planning and erecting these elaborate structures could not 
handle a problem which would rate today as a routine 
exercise for a high-school sophomore? 

The chroniclers of the tale offered no answers to these and 
similar questions. They confined themselves to the claim that 
Thales had solved the problem in a brilliant and rigorous 


fashion, by measuring the shadow of the Pyramid at an hour when a 
marfs shadow was equal to his height. 

The etymology of the word "pyramid" lends some authenticity 
to the preceding tale. The Greek pyramis was an adaptation of 
the Egyptian pyremus which,, curiously enough, denoted neither 
an imperial tomb nor a geometrical solid. Pyremus was Egyptian 
for altitude. The word was probably frequently used in extolling 
the loftiness of these monuments, and this might have led the 
Greeks to identify the word with the edifice itself, and later 
with any solid which resembled such an edifice in form. Be this 
as it may, the term eventually acquired an even broader signi- 
ficance. For, while all Egyptian Pyramids have square bases and 
congruent faces, the term "pyramid," as defined by Euclid and 
as accepted by us today, applies to any solid, symmetric or 
otherwise, with a polygonal base and triangular faces converging to a 

Now, we are concerned here not with the authenticity of this 
tale but rather with its plausibility, which is quite another 
matter. Indeed, it would be idle to speculate whether the 
Egyptians had actually propounded the problem to Thales, or 
whether he had solved the problem proposed. On the other 
hand, the answer to the following two questions would con- 
tribute materially to an appraisal of the status of mathematics 
of that period : First, was it within the ken of the Egyptian 
scholars of the period to determine the altitude of a pyramid? 
Second, could Thales have determined the altitude of the 
Great Pyramid by the method attributed to him, utilizing only 
such ideas as lay within his own ken? 

Consider a symmetric pyramid with a square base, PQRS, such 
as is shown in Figure i . Denote by a the side of the square ; by 
b the slant height, i.e., the perpendicular dropped from the apex 



A onto any one of the sides; by h the altitude of the pyramid. 
Both a and b can be measured directly, and will, therefore, be 
assumed given; the problem is to calculate h. We consider a 
vertical plane through the apex A of the pyramid and parallel 


to one of the sides of the base; it cuts the pyramid along the 
isosceles triangle ABC, the sides of which are <z, b and b respec- 
tively, and the altitude of which, AH, is the quantity h sought. 
The application of the hypotenuse theorem to the right triangle 
AHB yields directly: 

A -V* 1 -(frO 1 - 

Still, however simple this approach may appear today, it 
lay beyond the mathematical ken of the period under con- 
sideration. To be sure, the Egyptians knew of isolated cases 
where the hypotenuse relation held, such as the "triples" 
(3, 4; 5) or (5, 12; 13), but there is no evidence whatever that 
they were aware of the general validity of the Pythagorean 
theorem. Furthermore, a practical application of the theorem 


involves rational approximations to quadratic surds, a technique 
which, as far as we know, the Egyptians did not possess. 

This does not mean that the Egyptians had no means at their 
disposal for calculating the altitude of the Great Pyramid. 
The form of a symmetric pyramid with square base depends 
only on the ratio (k h/a) of altitude to the side of the base, 
which, in turn, completely determines the dihedral angle 
between the base and any one of the faces. Now, it so happens 
that these data are sensibly the same for all the Pyramids of 
Giza, the angle varying between 50 and 52, and the ratio 
between 0*63 and 0-64, averaging about 7/11. In other words, 
the Pyramids of Giza are very nearly similar solids, which 
means that if the ratio k were known for any one of the Pyra- 
mids, it could serve to calculate the altitude for any other. 
One of the smaller Pyramids the altitude of which was amenable 
to direct measurement could have been used to determine the 
ratio; or, and this is far more likely, a miniature model could 
have been built for the purpose at hand. And we must remember 
that the Egyptians were past masters of the miniature. 

This near-similarity of Egyptian Pyramids has been the subject 
of much speculation in recent times. One could explain this 
uniformity by observing that the construction of this imposing 
array of tombs was spread over more than a thousand years, so 
that the builders of the later pyramids had before their eyes 
magnificent models consecrated by tradition. But then why 
was the first pyramid erected on such odd proportions? Why 
these unorthodox angles and ratios? 

One plausible conjecture is that the choice was dictated by 
engineering considerations. The granite veneer of these tombs 
was laid upon massive masonry; to haul the bricks, mortar 
and stones up the steep slopes required the toil, tears, sweat 
and blood of an army of slaves. There was a limiting angle 
beyond which the expenditure in slave lives became uneco- 
nomical, for even the life of a slave had a price. Thus, the ratio 

D 49 


k might have been a sort of "coefficient of human endurance," 
and 7 /i i measured the limit of that endurance. 

However, such a matter-of-fact account would hardly satisfy 
those who have a penchant toward the occult. Indeed, nothing 
short of a religious, or at least aesthetic, interpretation is 
acceptable: the proportion must have been a sacrament, a 
criterion of beauty, or both! But why should 7 and u be 
singled out for this unique mission? Is it because of the singular 
role these magic numbers play in the vicissitudes of a game of 
dice? This is one occult argument that has not yet been advanced 
to explain the peculiarities in the design of the Egyptian 
Pyramids ; I hasten to add that some of the theories advanced 
are just about as reasonable. 

Most popular among the aesthetes is the theory which 
associates the design of the Egyptian Pyramids and every 
other design, human or divine with the so-called golden section. 
I shall deal in detail with these pretensions in the next chapter. 
The propounders of another theory point out that 11/7 is an 
approximation to %n, i.e., to the ratio of the semicircumference 
of a circle to its diameter. They claim that the designers of the 
Pyramids had chosen this proportion because they viewed the 
semicircle as a figure unexcelled in beauty. However, studies 
of the hieroglyphic papyri have failed to reveal any such 

"He determined the height of the Great Pyramid by measuring 
the shadow it cast at an hour when a man's shadow was equal 
to his height." In these words does the Greek historian 
Hieronymus describe the feat of the Wizard of Miletus. Other 
historians, classical and modern, too, reiterate this statement 
with some variations, but without critical comment. One is 
reminded of d'Alembert's bon mot: "It is not that history 
repeats itself, but that historians repeat each other." 

It would be simple enough to reconstruct the alleged solution 
of Thales if the problem had been to determine the altitude of 
an obelisk, or of any solid whose horizontal dimensions were 



negligible as compared with the vertical; for, at the time of the 
day when the sunrays struck the ground at an angle of 45, the 
shadow of the obelisk would be sensibly equal to the height, 
and such inaccuracy as would be introduced by the horizontal 
dimensions of the solid could be readily discounted. But the 
object was not an obelisk; it was a massive pyramid, the hori- 
zontal and vertical dimensions of which were of comparable 

The shadow of a pyramid is a triangle, and the shape of the 
triangle depends not only on the relative dimensions of the 


solid and on the hour of observation but on the latitude of the 
place and on the orientation of the sides. Now, the Egyptians 
were sun worshippers, and it was the obvious intention of the 
builders to orient the horizontal edges of these tombs East- West 
and South-North. In this they succeeded admirably. The 
angular deviation of the South-North edges from the true 
meridian, the so-called azimuth, nowhere exceeds one fourth 
of a degree; as a matter of fact, recent surveying has shown 
that the azimuth of the Great Pyramid is less than 4 minutes. 
Thus, for all intents and purposes, we may consider the edges 
of the Great Pyramid as oriented along the cardinal directions 
of the compass. 

In Figure 2 the shadow triangle is PQT, and at the hour when 


the rays of the sun strike the ground at an angle of 45 the 
altitude of the pyramid and the line 777 are equal. If jPQjTwere 
an isosceles triangle, then to determine TH it would be suffi- 
cient to measure the perpendicular distance of the tip T to the 
edge PQ and add to it \a. However, the vertical plane which 
passes through the apex A perpendicular to the edge PQis not 
parallel to the ecliptic, i.e., to the plane of the apparent path 
of the sun but is inclined to it at a substantial angle. The effect 
of this "obliquity" is that the triangle PQT is not isosceles, 
and this materially complicates the computation of the line 
TH which measures the altitude of the pyramid. 

As a matter of fact, in order to determine the distance between 
the tip T of the shadow and the centre H of the base of the 
pyramid, it would be necessary not only to measure the sides 
of the triangle PQT but also to carry out calculations involving 
repeated application of the Pythagorean theorem, and cul- 
minating in the extraction of a square root, calculations which 
by far transcend the ken of the period under consideration. 
On the other hand, we cannot rule out the possibility that 
Thales turned the difficulty by some artifice based on the 
principle of similitude. 

One such artifice is shown in Figure 3. Let HA and ha indicate 
two vertical posts, and HT and ht their shadows at some given 
time; these shadows are proportional to the altitudes of the posts. 
Suppose next that at another time of the day the respective 
shadows of the two posts are HT' and ht'\ these, too, are pro- 
portional to the altitudes. It follows that ft : th = T'T: TH. 
In particular, if ft = th, then T'T = TH. 

Returning now to the situation which allegedly confronted 
Thales, let ha designate the position of the man whom he was 
observing, and let ht be the shadow of the man at the time of 
the afternoon when one's shadow equals one's height; finally, 
let T be the tip of the shadow cast by the Pyramid at that 
time. Thales marks the points T and t, and with t as centre. 





draws the circle which passes through h. He then waits; as the 
shadows lengthen, there comes a time when the tip of the man's 
shadow strikes the circle drawn, say in t'. Simultaneously, 
the tip of the Pyramid's shadow has moved from T to T', and 
7T' = TH = HA 9 because tt' = th = ha. Thus the altitude 
of the point A above the plane of the base is determined by 
the horizontal line TT', which is fully accessible to direct 

This conjectured solution depends neither on the latitude of 
the place, nor on the obliquity of the ecliptic, nor, for that 
matter, on the form of the solid. It would obviously apply to 
determining the altitude of any point A, provided two consecu- 
tive positions of its shadow T and T' were known. Besides, it has 



the merit of presupposing only such ideas and methods as 
existed at the time when the incident is alleged to have occurred. 
In short, what the conjecture lacks in authenticity, it makes up 
in historical plausibility. 

Indeed, the conjecture would be plausible even if the epi- 
sode had taken place at a much earlier age, since the conception 
of similitude antedates deductive geometry by thousands of 
years. For, not only is similitude a prerequisite to all geometrical 
thinking, but it dominates the graphic arts as well, and the fact 
that these arts have been cultivated since time immemorial 
shows how deeply the conception is rooted in man's con- 


Chapter Five 

I shall indulge my sacred fury .... 


In one of the Dialogues, Plato puts into the mouth of Timaeus, 
a follower of Pythagoras, the following words : "It is impossible 
to join two things in a beautiful manner without a third being 
present, for a bond must exist to unite them, and this is best 
achieved by a proportion. For, if of three magnitudes the mean 
is to the least as the greatest to the mean, and, conversely, 
the least is to the mean as the mean to the greatest then is 
the last the first and the mean, and the mean the first and the 
last. Thus are all by necessity the same, and since they are the 
same, they are but one." 

The problem to which this exotic verbiage alludes has come 
to be known as golden section, or division of a magnitude into 
extreme and mean reason, the word "reason" being used here in 
the archaic sense of ratio. Translated into mathematical 
language the problem is to divide a given magnitude, say a 
rectilinear segment of length s, into two parts, such that the 
greater, x, be to the whole as the lesser part is to the greater. 
Hence the proportion: 

x : s = (s x) :x. . . . (5.1) 
This, in turn, leads to the quadratic equation, 

x* + sx-s*=o, . . . (5.2) 
the positive root of which is 

* = MV5-0- (5-3) 




A simple straightedge-compass construction, based on a 
direct interpretation of this last formula, is shown in Figure 4. 
In the right triangle ABF we have by construction: AB = s, 
SF = j = fiF and J^ = 45 1 . Hence : 

i.e., the point 

divides the given segment ;!# in the ratio 

In the Dark and Middle Ages, the golden section became a 
favourite topic of theological speculation. Many a Schoolman, 
inspired by the arguments of the Pythagoreans and Platonists, 
sought and found in the proportion a key to the mystery of 
creation, declaring that extreme and mean reason was the very 
principle which the Supreme Architect had adopted in cosmic 
and global design: hence, the title "divine proportion" bestowed 
upon the ratio. Nor were these mystic meditations the monopoly 
of medieval monks. The virus affected quite a few poets and 
painters of the Renaissance, including Leonardo da Vinci 
himself. Of this, however, later. 

The modern revival of the golden-section cult, like so many 
other movements of the kind, is characterized by what may 
be called * 'rationalizing the occult." Its devotees, mostly 



artists, accentuate the aesthetic value of the proportion, its 
prevalence in nature, the exhaustive role it plays in human 
anatomy, its cosmic significance. They claim that the golden 
section is the clue to the beauty of Greek sculpture, as well as 
to the finest specimens of antique architecture such as the 
Egyptian Pyramids. 

One of these claims is illustrated in Figure 5. The sides of 
the rectangle are in "divine proportion." The pattern will 


strike the reader as quite familiar, since many objects in our 
immediate environment have this particular design: windows, 
tables, books, boxes, playing cards. We are assured by golden- 
section enthusiasts that this pattern is the incarnation of 
grace, and that it is recognized by most people as such. Are 
these contentions supported by facts? Well, as far back as 
1876, the German psychologist, Gustav Theodor Fechner, 
spurred on perhaps by the golden sectionists of his day, con- 
ducted a series of experiments on a large and heterogeneous 
group of people. Ten rectangular patterns ranging in form 
from 2 by 5 to i by i, and including the golden section, were 
placed at random in a room, and the individuals were requested 



to record as to which of these shapes appeared to them as the 
most graceful. While the golden section received more votes 
than any other, it appealed only to one third of those inter- 
viewed. On the whole, the results were rather inconclusive. 

Among the attempts at accounting for this alleged preference 
on rational grounds is the contention that when the human 
eye surveys a rectangular pattern, it instinctively separates 
from it a square. (Figure 5.) Ostensibly, the closer the remain- 
der resembles the whole, the more the pattern appeals to the 
eye. The ideal, of course, is provided by the golden-section 
rectangle, inasmuch as here the residual rectangle is similar to 
the original. Such endeavours to trace one doubtful predilection 
to another just as doubtful are characteristic of the specious 
reasoning of the aesthete. 

In appraising the validity of these claims, one should bear in 
mind that the golden-section "ratio" is an irrational number. 
Indeed, setting in equation (5.2) x/s = Z), we find 

D* + D-i=o, . . . (5.4) 
the positive root of which is 

=KV5-i)- (5-5) 

The numerical value of D with five correct decimal places is 
0-61803. To vindicate their claims that the golden section is 
the motive of such and such a design, human or divine, the 
golden sectionist will seek to prove that the measured value 
agrees with the one calculated, within "reasonable limits," 
of course. Obviously, the success of his undertaking will depend 
on the latitude allowed in interpreting the term "reasonable 

A case in point is the allegation that the designers of the 
Egyptian tombs had been guided by the golden-section rule. As 
mentioned in the preceding chapter, the ratio of altitude to 
side has an average value of 0-625, an d never falls below 



o 63 for any of the Pyramids of Giza. The mean discrepancy 
between this ratio and the extreme and mean reason is more 
than 2 1 per cent., and such a difference could hardly be 
ignored without stretching reason to the extreme. 

Just as specious are the contentions that certain proportions 
in human anatomy conform to the golden-section rule, as for 
example that the navel divides the average man's height in 
extreme and mean reason. Even if such biometrical data could 
be established, how could one reconcile them with the aesthetic 
claims mentioned above? For, surely, even the most ardent 
golden sectionist would hardly contend that the average 
human stature embodied his idea of grace. 

Now, given the privilege of selecting the traits to be com- 
pared, the freedom of choosing and grouping the specimens 
which are to be measured for these traits and a latitude in the 
degree of precision allowed in interpreting the data measured 
with all these liberties at one's disposal, one should be able to 
turn any recondite mystery into a mathematical law, and, as 
a matter of fact, into a law assigned in advance. This sounds 
like accentuating the obvious, and so it is. But, evidently, it is 
not so obvious to the authors of some of the studies, biometric, 
econometric, psychometric, which have come to my notice. 

While on the subject of rational approximations to irrational 
magnitudes, I must mention one striking property of the extreme 
and mean reason which could have furnished much grist to 
the occult mill of the golden sectionists, if the medium in 
which this property is expressed lay within the ken of the 
cult; I am speaking of the expansion of the irrational \(<J 5 1) 
into a continued fraction.* 

I shall have much to say about this important device later. 
For the purpose at hand it is best to approach the matter from 
the heuristic point of view. We have by definition 

D : i = i : (i + D). 

* See Chapter 12. 



Accordingly, we can write 

from which we infer that the golden-section ratio is the limit 
of the infinite continued fraction 

I + - 

I+ 7T7.. . (5 .6) 

the most simple of its kind, since not only are all the denomi- 
nators equal, but their common value is one. 

To the mystic, the mere circumstance that an entity can be 
expressed by a single symbol is portentous enough; but when 
that one symbol is one, then the divine origin of the entity 
transcends all doubt. For, one is the emblem of God; in the 
words of the mystic Leibnitz: "One has sufficed to draw all 
out of nought." That it would require an infinitude of steps to 
attain the desired goal lends power to the interpretation, inas- 
much as the infinite, too, is an attribute of the Deity. Finally, 
there is an absolute quality to a continued fraction which no 
other representation possesses : it is independent of the scale of 
numeration. Thus, the "spectrum" would have been the same 
had Providence chosen to equip man with 12, or 60, or any 
number of fingers in lieu of the random 10. This disquisition 
is offered here for what it is worth as this author's humble 
contribution to the cult of the occult. 

A favourite method of exhibiting the alleged role played by 
the golden section in human anatomy is shown in Figure 6. 
The picture is taken from a modern book on the subject, but 
the scheme of presenting the posture of a man as a five-pointed 
figure can be traced to Leonardo da Vinci and his mathe- 




matical collaborator, the monk Luca Pacioli. The latter 's 
book, entitled Divine Proportion, contained a number of striking 
drawings by da Vinci, two of which are reproduced in Figure 9. 
The tract was published in 1509, and many of the ideas pro- 
pounded by the modern devotees of the cult hark back to that 

The point of departure of these occult speculations is an 
isosceles triangle with angles 36, 72 and 72 (ABC in Figure 
7). In such a ' 'golden-section triangle," the angular bisector, 
AD, determines two isosceles triangles., DAB and DAC, and the 
latter is similar to the original triangle. From this follows : first, 
that the point D divides the side EC in extreme and mean reason; 
and second^ that the sides of a golden-section triangle are in divine 





The configurations affected by these lemmas are: 

(a) The regular pentagon. (Figure 8a.) Here, ACD is a golden- 
section triangle ; hence, the side and diagonal of a regular pentagon 
are in divine proportion. 

(b) The regular pentagram. (Figure 8b.) Here ABC is a golden- 
section triangle. It follows that the sides of a regular pentagram 
divide one another in golden section. 

(c) The regular decagon. (Figure 8c.) Here AOB is a golden- 
section triangle; hence, the side of a regular decagon and the radius 
of the circumscribed circle are in divine proportion. 





While the origin of the golden-section idea is obscure, it is 
quite probable that the Pythagoreans were led to endow the 
ratio with occult significance because of the property of the 
pentagram described above, namely, that ite sides divide one 
another in extreme and mean reason. We know that the 
pentagram had played an important role in the ritual of many 
ancient people, that it was a sacred symbol of the Pythagorean 
Order, and that to this day some "secret" societies which hail 
Pythagoras as their spiritual forebear use this mystic figure as 

The effect of Christianity on popular fancy was to turn the 
sacred into the occult. The pentagram of the Greeks became 
pentacle 9 indispensable item of the sorcerer's gear. In some 
places the mark of the pentacle was a tiding of evil; in others, 
on the contrary, it was viewed as a sure deterrent against 
Satan's machinations. In some languages it was called deviVs 
hoof y in others witcKes foot. In the course of time, the geo- 
metrical origin of the term was all but forgotten, until pentacle 
became a symbol of black magic and the conjuror's art. 

Several theories have been advanced to explain this strange 
predilection. One of these traces the preference to the five 
fingers of the human hand, maintaining that the occult powers 
ascribed to the pentagram did not derive from its geometrical 
properties, but from its association with the number Jive. It 
may be pointed out in this connection that the Platonists 
attached as much importance to the pentagon as to the penta- 
gram, and that the number five was an important adjunct to 
their cosmic speculations. 

This brings me to another corner of that occult fringe which 
surrounds the early history of geometry. 

Consider first a regular convex polygon of n sides. The n 
vertices of the figure lie on a circle, and there exists another 



circle which is tangent to all n sides. Thus the construction of 
a regular n-gon is fully equivalent to the division of the circum- 
ference of a circle into n equal arcs, a problem, known as 
cydotomy. The problem admits of a solution for any value of 
the integer n, and however involved may be the actual con- 
struction of the corresponding n-gon, the difficulty is technical 
in nature and not a matter of principle. 

When we pass from the plane to space, we find the situation 
quite different. In lieu of polygons we have here polyhedra, i.e., 
solids bounded by planes. The boundary planes of a poly- 
hedron intersect in lines which are called edges \ the edges, in 
turn, converge in the vertices of the solid and combine into 
polygons which are called the faces of the polyhedron. Denote 
the number of faces, edges and vertices of any given poly- 
hedron by/, e and v respectively: these integers are connected 
by certain relations known as Euler equations. 

When we say that a polyhedron is convex we mean that the 
solid lies on one side of any one of its faces; when we say that the 
polyhedron is regular, we mean that its faces are congruent 
regular polygons. When both of these conditions are met, then 
the vertices of the solid lie on a sphere and there exists another 
sphere which touches all the faces of the solid. Let n be the 
number of sides in any face, and m the number of edges which 
emerge from any vertex: then the Euler equations mentioned 
above reduce to 


Five solutions of these equations are given in the table below, 
and it is not difficult to show that no others exist. 








Tetrahedron . 
Hexahedron (Cube) 









Octahedron . 














Icosahedron . 










Two of these polyhedra are shown in Figure 9. They are taken 
from the Divine Proportion of Pacioli, who attributed the draw- 
ings to Leonardo da Vinci. The choice was obviously motivated 
by the fact that both solids contain regular pentagons, the dodeca- 
hedron being distinguished by its pentagonal faces, while the 
five triangles which emanate from a vertex of the icosahedron 
form a pyramid with pentagonal base. 


The discovery of the regular polyhedra has been ascribed to 
Plato; hence the name of "Platonic solids" under which these 
bodies are commonly known. That Plato was their discoverer 
may be seriously doubted, but the knowledge that there were 
five such solids, and only five, must have been very gratifying 
to him and his disciples. However, this posed a new problem: 
cosmic harmony demanded a one-to-one correspondence 
between these perfect solids and the basic constituents of 
matter, and Platonic cosmogony recognized only four such 
primordial elements: earth, fire, water and air. 

The vexing problem was eventually solved by invoking a 
principle which has governed occult speculations since time 
immemorial: "When in doubt, let the Deity have the hind- 
most!" One of the perfect solids must be assigned to the 
heavens! But which? The most perfect, of course : the dodeca- 

E 65 


hedron, whose pentagonal faces bore the imprint of the perfect 

And so it was that having assigned the cube to the firm 
earth, and having conferred a solid each on the more epheme- 
rous fire, water and air, Plato dedicated the dodecahedron 
with its sacred pentagonal faces to the heavens. 

I shall now quote from a letter written two thousand odd years 
later: ". . . Before the universe was created, there were no 
numbers except the Trinity which is God himself . . . For, the 
line and the plane imply no numbers: here infinitude itself 
reigns. Let us consider, therefore, the solids. We must first 
eliminate the irregular solids, because we are concerned here 
only with orderly creation. There remain six bodies, the sphere 
and the five regular polyhedra. To the sphere corresponds the 
outer heaven. For, the universe is twofold: dynamic and static. 
The static is the image of God-Essence, while the dynamic is 
but the image of God- Creator, and is therefore of a lower 
order. In its very nature, the round corresponds to God and 
the flat to his creation. Indeed, the sphere is threefold : surface, 
centre, volume; so is the static world: firmament, sun, ether; 
and so is God: Son, Father, Spirit. On the other hand, the 
dynamic world is represented by the flat-faced solids. Of these 
there are five; when viewed as boundaries, however, these five 
determine six distinct things ; hence, the six planets that revolve 
about the sun. This is also the reason why there are but six 
planets. And because the sun stands at the centre of creation, 
and because it is at rest and yet the source of all motion, it is 
the true image of God, the Father, the Creator. For, what 
God is to creation, is motion to the sun ..." 

". . . I have further shown that the regular solids fall into 
two groups: three in one, and two in the other. To the larger 
group belongs, first of all, the Cube, then the Pyramid, and 
finally the Dodecahedron. To the second group belongs, first, 
the Octahedron, and, second, the Icosahedron. That is why 
the most important portion of the universe, the Earth 



where God's image is reflected in man separates the two 
groups. For, as I have proved next, the solids of the first 
group must lie beyond the earth's orbit, and those of the 
second group within. . . . Thus was I led to assign the CUBE to 

MERCURY. . . . 

Does this sound to you like the raving of a maniac, or a 
page out of Madame Blavatzky? Then be reassured ; these are 
excerpts from Johannes Kepler's own preview of a tract which 
he published in 1596 under the title The Cosmic Mystery. And 
such was the spirit of that period that when the essay reached 
Galileo and Tycho Brahe, both astronomers responded with 
flattering comments. As a matter of fact, the latter was so 
impressed that he invited young Kepler to become his assistant, 
urging him at the same time to apply his mystic methods to 
the Tychonian cosmic system which was sort of a cross between 
the Ptolemaic and the Copernican, the planets spinning 
around the Sun, while the latter was executing an exotic 
pirouette about the Earth. 


Kepler called his correspondence between planets and poly- 
hedra Mysterium Cosmographicum. That he regarded it not as 
mere rhetorics but as a pictorial presentation of an actual 
mathematical relation which governed the solar system is 
attested by the sketch reproduced in Figure 10. To be sure, the 
projected model never did get beyond the drawing stage; 
not, however, because Kepler had lost faith in his youthful 
conception, but because he could not raise the funds for its 

Now, there is nothing loose or even indeterminate in the 
mathematics of the scheme; if anything, it is too rigid. Indeed, 
if we assume, as Kepler did, that the planetary orbits are con- 
centric and coplanar circles with the sun in the centre of the system, 



then the first step is to determine the relative magnitudes of 
the radii of these circles. Denote these radii by the initial 
letters in the names of the six planets, s, 7, m, e, v and m'\ 
and let S, J, M, E, V, M' designate the six spheres which admit 
these orbits for great circles. Then, according to Kepler, there 
exists a cube the vertices of which rest on the sphere S, and 
the faces of which are tangent to the sphere J. Since the diagonal 
of that cube is the diameter of S, and its side the diameter of 
jf, it follows that s is to j as the diagonal of a cube is to its side, 
i.e., s/j = ^3. By similar geometrical considerations one can 
derive the ratios m/j, e/m, etc., culminating in a set of propor- 
tions: a complete arithmetic counterpart of the Mysterium 
Cosmo graphicum . 

The question is to what extent do these results agree with 
the subsequent observations of Kepler and with the laws 
which he deduced from these observations? Now, the first of 
these laws declares that the planetary orbits are not circles but 
ellipses with the sun as common focus ; well, the eccentricities of these 
orbits are so small that they could readily be taken for circles 
as a first approximation. The second law declares that the 
motion of a planet is such that equal sectors are covered in equal 



intervals of time) while the third states that the square of the 
period consumed in a complete revolution is proportional to 
the cube of the major axis of the planet's orbit. Both the second 
and the third law hold for circular orbits. Thus, with certain 
reasonable reservations, the Mysterium could be accepted as 
the model of the solar system, provided that the proportions of 
which I spoke above were in agreement with astronomical data. 

The puzzling fact, however, is that these proportions do 
not even remotely agree with those derived from observations. 
Thus we found s/j = ^3 = 1*732, whereas the ratio of the 
mean distances to the sun of Saturn and Jupiter is i 833 . . . 
The error, of nearly 6 per cent., is of a size that cannot be 
taken lightly, and the discrepancy between observed and 
calculated values is even greater for the remaining propor- 
tions. It has been argued that the Mysterium was designed 
before Kepler got access to Tycho Brahe's observations, and 
this is certainly true. Still, Kepler never did retract his early 
work; on the contrary, a quarter of a century later, in a book 
entitled De Harmonici Mundi, he amplified the Mysterium by 
endowing it with sound: under the impact of the moving 
orbs, the invisible spheres emitted tones of varying intensity 
but of pitch so high that only the sentient soul of God, who 
dwelt in the sun, could perceive this music of the spheres. 


There has been a tendency on the part of some of Kepler's 
biographers to palliate his occult activities and to portray the 
man as a detached observer bent on ascertaining the truth no 
matter how sharply this truth contradicted his own precon- 
ceptions. These interpreters point out that Kepler's title of 
Imperial Astronomer carried more honour than honorarium; 
that even this scant pay was perennially in arrear, so much so 
that upon his death the exchequer still owed him 20,000 
florins; that to supplement this meagre income Kepler had to 
resort to reading horoscopes, which, allegedly, explains his 
extensive excursions into astrology. As to his mystic specula- 
tions on the nature of the cosmos, these interpreters hint that 



Kepler had uttered them with tongue in cheek, largely as a 
concession to the spirit of his time, or, at worst, as a means 
to impress the half-educated men on whom his advancement 

Now, unlike the mystic philosophers of classical Greece, 
Kepler left behind a wealth of autobiographical material. 
This Kepleriana suggests to my mind not the portrait of a 
learned opportunist but that of a mystic obsessed with the 
conviction that he had been chosen by Providence to reveal 
to man the essential unity between the motions of heavenly 
bodies, the harmony of sound, the logic of number and the 
beauty of geometrical form. It was this conviction, this "sacred 
fury" as he called it, that impelled him, in my opinion, to 
pursue, throughout life and to the end, the cosmic phantom 
which he had conceived in his youth. He wrote: "Nothing 
shall stop me. I shall indulge my sacred fury. I shall triumph 
over mankind. I have stolen the golden vases of the Egyptians 
to erect a tabernacle to my God. . . . The die is cast, the book 
is written to be read now or later, I care not which. It may 
well wait a century for a reader: has not God waited for an 
observer these six thousand years?" 

Was he referring to his laws of planetary motion or to his 
Mysterium Cosmograpkicum? Well, as far as we know he made 
no such distinction. Both were parts of the sacred tabernacle, 
and I am quite sure that, faced with the choice, he would 
have put the Mysterium first. 

No, the arguments of the apologists notwithstanding, I, for 
one, cannot escape the conclusion that it was while endeavour- 
ing to substantiate his occult illusions that he had discovered 
the laws without which Newton could not have, in turn, 
discovered the principle of universal gravitation. 


What could impel men of unquestioned scientific com- 
petence and integrity to carry out protracted and painstaking 
observations, subject these data to a keen and thorough 



mathematical treatment, and at the same time profess to see 
in their work the confirmation of some phantastic code which 
they have imposed on God or Nature? I have no answer to 
this question, but I have lived long enough to know that this 
schizophrenia, so manifest in Kepler, is far more prevalent 
among men of science than is commonly believed. 

The occult has had many facets, and not all of these were 
of the naive variety described in this chapter. In mathematics, 
the occult never outgrew these cruder forms, and that may be 
the reason why here it remained on the fringes of the science. 
In some fields contiguous to mathematics the occult has 
assumed more subtle and hence less recognizable forms. 
Indeed, listening to some modern physical theories one is 
often at a loss to understand wherein these exotic speculations 
differ from the mystic introspections of a Pythagoras or a 

The occult was not born with Pythagoras, nor has it died 
with Kepler. The historian looks back and beholds that much 
of the sacred of a yesteryear is condemned today as occult, 
and he wonders what will become of our discarded axioms. 



. . . such folly, unfortunately, is never confined to 
one subject, since the habit of fallacious thinking, even 
as that of correct reasoning, has a tendency to increase. 


Among the imposing array of problems which have graced 
mathematics during its long and eventful history there is a 
small group which has exerted a peculiar fascination on the 
amateur. To this group belong the famous problems of con- 
struction on which I touched in earlier chapters; here also 
belongs the Euclidean postulate of parallels. Several other 
problems may be listed under this head; of these, I shall 
mention only the Fermat problem: to demonstrate that the 
equation x n + y n = z n has no solutions in whole numbers as 
long as n itself is an integer greater than 2. 

While the difficulties inherent in these problems can only be 
met by the resources of modern algebra and analysis, their 
preliminary formulation requires neither the exotic symbolism 
nor the weird terminology which obscure to the layman so 
many mathematical questions. Thus, whether the issue is con- 
struction or proof, the aims of these problems appear clear 
cut and direct, even to those who possess little mathematical 
training or insight. This tantalizing simplicity has deceived 
many an amateur, with the result that no end of "solutions" 
appeared, one as pretentious and as preposterous as the other. 

While the amateur receives but little encouragement from 
the orthodox mathematician a circumstance which he 
promptly attributes to professional jealousy he finds the news- 
paper editor quite receptive, as a rule. In fact, some papers have 
regarded such achievements of sufficient importance as to be 



featured as front-page news; others even have gone so far as 
to herald in glaring headlines that such or such an age-old 
problem,, which has defied the efforts of countless generations 
of professional mathematicians, has in this day and age been 
completely solved by a nonprofessional. In this manner the 
amateur may gain considerable notoriety; unfortunately, his 
glory is soon eclipsed when a competing amateur with a con- 
flicting solution of the same age-old problem manages to gain 
the ear of another equally credulous editor. It is by dint of 
such publicity that the "famous" problems are daily gaining 
in fame to the amusement of the expert and the confusion of 
the public. 

These problems have played a considerable role in the history 
of mathematics. Not that they were key problems, in the sense 
that without their exhaustive solutions the disciplines in which 
they had arisen could not have progressed. No, their role can 
best be compared to that of a catalytic agent which precipitates 
chemical action without participating in it. Thus, these prob- 
lems have been responsible for the invention of many a new 
method, and, more often than not, whole new disciplines have 
followed in their wake. As in the parable of the vineyard, the 
heirs had failed to unearth the gold after laboriously plowing 
up the estate; yet, the loosened soil yielded a harvest which 
exceeded in wealth the anticipated treasure. 

The ancient problems of trisection of an angle and duplica- 
tion of a cube have in modern times led to the theory of 
equations and have been indirectly responsible for the intro- 
duction of the exceedingly important concept of group. The 
attempts to square the circle led to the discovery of transcen- 
dentals; the efforts to prove Fermat's theorem resulted in the 
theory of ideals; the failures to demonstrate the postulate of 
parallels culminated in the discovery of the non-Euclidean geo- 
metries, without which the theory of relativity would be 

Of all these developments the ambitious amateur is, of course, 



wholly unaware. Indeed, he is, as a rule, not interested in de- 
velopments which require the study of the work of others. 
Sufficient unto him is to knoV that the problem which he tackles 
has not yet been solved, or that it has been declared impossible 
by professional mathematicians. For the rest, he trusts in God 
and in his own powers or prowess. 

It has fallen to my lot to come in contact with many of these 
individuals. I say this in no spirit of complaint, for whatever 
inconvenience or irritation they may have caused me has been 
amply rewarded by the experience which I gained while 
studying their unusual turn of mind. Indeed, I hoped at one 
time to use the accumulated material in an essay on the pathology 
of human reasoning. As years go by, however, it is becoming 
increasingly doubtful that I shall ever have the leisure to 
engage in such a project. Accordingly, I decided to present 
some of the material here. 

While the mental kinks of these individuals are to mathe- 
maticians little more than objects of passing curiosity, the same 
should not be true of psychiatrists, more particularly of those 
who specialize in the study of megalomania. Indeed, I cherish 
the hope that among my readers there may be some such 
specialist who, stimulated by these casual remarks, will under- 
take a scientific investigation in this field which, to my knowl- 
edge, has not even been touched by the alienist. 

It will be convenient to designate this specimen of humanity 
by a special name. The term "circle-squarer" is obviously 
misleading, for rarely do these individuals confine their efforts 
to this one classical problem. More often than not, they will 
regard any mathematical problem or any problem, for that 
matter as particularly adapted to their talents, provided the 



experts have failed to solve it or have reached negative results 
in its regard. 

I propose to call these persons pseudomaths, a term coined by 
Augustus De Morgan. A substantial portion of his Budget of 
Paradoxes was devoted to the study of these individuals and their 
fallacies. Of the obsession with which the species is afflicted 
De Morgan had this to say: "The pseudomath is a person who 
handles mathematics as the monkey handled the razor. The 
creature tried to shave himself, as he had seen his master do; 
but not having any notion of the angle at which the razor was 
to be held, he cut his own throat. He never tried again, poor 
animal! But the pseudomath keeps to his work, proclaims him- 
self cleanshaved, and all the rest of the world hairy. . . . The 
feeling which tempts him to these problems is that which, in 
romance, made it impossible for a knight to pass a castle which 
belonged to a giant or an enchanter. This rinderpest of geo- 
metry cannot be cured when once it has seated itself in the 
system. All that can be done is to apply what the learned call 
prophylactics to those who are yet sound. When once the virus 
gets into the brain, the victim goes round the flame like a 
moth first one way and then the other, beginning again where 
he ended, and ending where he began." 

They come from all strata of society and all walks of life. While 
the male of the species predominates, ladies, too, have entered 
the race. In fact, I have noticed of late that the number of 
feminine pseudomaths is on the increase which, perhaps, is but a 
symptom of the gradual emancipation of the fair sex. Most 
countries, races, creeds, professions and crafts are represented; 
my own list includes farmers and army officers; bankers, 
brokers and merchants; realtors and prospectors; doctors, 
dentists and lawyers; engineers, artists and artisans; teachers, 
preachers and even a college president. 

How large is their number is a question that cannot be 
answered with any degree of accuracy. There are no societies 
of pseudomaths, which is not surprising, since every pseudo- 



math, being in sole possession of the eternal truth, views every 
other as an impostor and a fraud. In the absence of such rosters, 
any estimate is but a guess; my own, based on personal con- 
tacts and correspondence, is that in this country alone their 
number must run into many thousands. 

In spite of the fact that they are being recruited from so 
many different occupations and classes, they exhibit a remark- 
able similarity in their methods of approach to a problem as 
well as in the strategy they use to obtain recognition. When 
given an opportunity to defend their contentions, every one of 
the pseudomaths who has come under my observation used the 
same tactics, which may be best qualified as a policy of attrition. 
With tiresome laboriousness and endless detail he would 
demonstrate the obvious steps in his reasoning; but, arrived at 
the critical point, he would pass over it with the utmost speed. 
Indeed, I have found in my dealings with them that I can save 
myself a great deal of ennui by just listening listlessly and with- 
out the slightest interruption to their drone, patiently awaiting 
the imminent slur. 

The variety of their interests is unbelievably great. Thus, the 
announcement in 1907 of the Wolfskehl prize for the first solu- 
tion of the Fermat problem found such a tremendous response 
from the pseudomaths of the world that the handling of the 
correspondence became a gigantic task. The advent of the 
theory of relativity has deviated the efforts of many into this 
new channel, with the result that every so often we are graced 
with a new refutation of Einstein. Quite often I receive letters 
from some individual who has discovered a kinship between 
phenomena which to the benighted scientist appear worlds apart, 
while one possessed by a truly universal spirit has succeeded in 
uniting into a single synthesis the Euclidean postulate of 
parallels and the quadrature of the circle, the Fermat problem 
and perpetual motion, the principle of relativity and the 
existence of the Deity, the quantum theory of the atom, fore- 
casts of the stock market, the abolition of wars, the solution 
of the econonic depression and the liberation of mankind from 
the Bolshevist scourge to mention but a few of the achieve- 
ments he claims. 


Their dogged perseverance defies all description. They seem to 
thrive on abuse, discouragement and ridicule. They speak of 
their undying devotion to truth. It must be conceded that no 
pseudomath has ever derived material benefit from his dis- 
coveries, while most of them continually sacrifice wealth and 
position in their efforts to gain recognition. I do not believe 
that they are actuated by greed; in fact, I have a lurking 
suspicion that even those pseudomaths who have tried for the 
Wolfskehl prize of 100,000 marks were motivated more by the 
desire to justify in the eyes of their kin and friends their fruitless 
efforts of many years than by any hope of winning the prize. 
To my mind the dominating motive which sustains them in 
the face of all failures is an inordinate craving for publicity. 
To attain this end they will stop before no humiliation. One 
of the most remarkable instances of this kind is the case of one 
James Smith, a merchant of Liverpool, who flourished in the 
sixties of the last century. Smith spent half of his life and a con- 
siderable fortune in defending his method of squaring the circle, 
a method which, with all irrelevancies removed, amounted to 
the declaration that the ratio of the circumference to the dia- 
meter of a circle was equal to exactly 25/8. He engaged in a 
voluminous correspondence with the leading British mathe- 
maticians of his time, among whom were such outstanding men 
as William Rowan Hamilton, Stokes, Clifford, and De Morgan. 
They all began by trying to set him right; they all ended by 
giving it up as a bad job. Some of the replies of these men were 
so devastating that any sane man would have immediately 
destroyed them for fear that they might see the light of day. 
Not James Smith! At a price to himself that must have 
amounted to a small fortune, he published the whole corre- 
spondence and distributed the book free of charge to friend and 
foe alike. This volume of five-hundred-odd pages is a human 
document of inestimable value to a psychopathologist. 



There is a term used in physics to designate an effect which 
persists after the generating cause has ceased to act, such, for 
instance, as occurs in elastic or magnetic phenomena. Such a 
residual after-effect is known as hysteresis and could be aptly 
applied to many phenomena in the history of science, and of 
culture more generally. 

The pseudomath is a case in point. Far from being a pheno- 
menon peculiar to our own times, he is as old as mathematics 
and, in a certain sense, even older. Indeed, just as astronomy 
was preceded by astrology, and chemistry by alchemy, so was 
mathematics preceded by pseudomathematics. Thus, in the pre- 
logical period of mathematics, all its adepts were pseudomaths, 
more or less. The deductive method has put an end to the 
usefulness of the pseudomath, yet he shall long persist as a sort 
of hysteresis. 

That this specimen was already a problem in ancient Greece, 
even as far back as the days of Pericles, may be judged from a 
scene which occurs in Aristophanes' comedy The Birds. Meton, 
an Athenian surveyor and a pseudomath if there ever was 
one demands admission to the Bird State. When requested 
to give his qualifications, he offers to parcel off the atmosphere 
into acres, and to square the circle by means of a straightedge. 
He is refused admittance and asked to move on. He insists on 
knowing why, whereupon the following conversation takes 
place: ''What danger is there? Is discord raging here?" "No, 
not at all !" "What is the matter then?" "In perfect concord are 
we resolved to kick out every humbug." It is interesting to note 
that twenty-four hundred years ago Aristophanes had a proper 
appreciation of such amateurish efforts, whereas editors of 
some of our modern dailies rarely miss a chance to blazon 
forth to the world that on such and such a date such and such 
a nonprofessional has at last solved a problem which has baffled 
professional mathematicians for nearly three thousand years. 

The Dark Ages may be viewed as a sort of resurrection of 
the prelogical period. What little science was cultivated then 
was so hopelessly mixed up with pseudoscientific ideas that the 



task of a modern historian who deals with that period may well 
be compared with that of unscrambling an omelet. Mathe- 
matics was no exception: the famous problems were approached 
in the same spirit as the search for the philosopher's stone, or 
for the elixir of life. In fact, it was held by many that the quad- 
rature of the circle would open the door to many such mysteries. 
Some of the most absurd solutions of the famous problems date 
from that period; furthermore, most of the fallacies of the 
modern pseudomath may be found in medieval literature. 
That there is little new under the sun applies to fallacies even 
more than it does to truth. 

With the advent of modern times, there was an unprecedented 
increase in pseudomathematical activity. During the eighteenth 
century all scientific academies of Europe saw themselves 
besieged by circle-squarers, trisectors, duplicators and per- 
petuum mobile designers, loudly clamouring for recognition of 
their epoch-making achievements. In the second half of that 
century the nuisance had become so unbearable that, one by 
one, the academies were forced to discontinue the examination 
of the proposed solutions. The first to inaugurate this policy 
was the French Academy. To its published resolution there was 
attached an explanatory note written by the great Condorcet. 
The following are excerpts from this interesting document: 

"The Academy has resolved this year not to examine in the 
future any solution of the problems of the doubling of the cube, 
the trisection of the angle and the squaring of the circle, or 
of any machine which lays claim to perpetual motion. . . . We 
have thought it to be our duty to account for the reasons which 
have led the Academy to adopt this decision. . . . An experience 
extending over more than seventy years has demonstrated that 
those who send in solutions of these problems understand 
neither their nature nor their difficulties, that none of the 
methods employed by them could ever lead to solutions of these 
problems, even were such solutions attainable. This long experi- 



ence has convinced the Academy of the little value that would 
accrue to science, were the examination of these pretended 
solutions to be continued." 

"There are still other considerations that have determined 
this decision. A popular rumour has it that the Government has 
promised considerable rewards to one who would first solve the 
problem of squaring the circle. . . . On the strength of this 
rumour, a multitude of people, much greater than is commonly 
believed, have given up useful work to devote their time to this 
problem which often they do not understand, and for which 
none of them possess the requisite preparation. Nothing could, 
therefore, serve better to discourage these people than this 
declaration of the Academy. Some of these individuals, being 
unfortunate enough to believe that they have been successful, 
have refused to listen to the criticism of geometers, often 
because they could not understand it, and have finished by 
accusing the examiners of envy and bad faith. . . ." 

"The folly of the Circle-Squarers would result in no greater 
inconvenience than the loss of their own time at the expense of 
their families, were it not that such folly is, unfortunately, 
never confined to one subject, since the habit of fallacious 
thinking, even as that of correct reasoning, has a tendency to 
increase, as it has happened in more than one case. Moreover, 
to account for the singular fact that without studying the 
subject they have arrived at solutions which the most famous 
scholars have vainly sought they persuade themselves that 
they are under the special protection of Providence, and from 
this there is but one step to the belief that any combination of 
ideas, however strange, that may occur to them are so many 
inspirations. Humane consideration therefore demanded that 
the Academy, persuaded of the uselessness of such examinations, 
should seek to offset by public announcement a popular opinion 
that has been detrimental to so many families. . . ." 

"Such were the principal reasons that have determined the 
Academy's decision. The declaration that it will not engage in 
the future in this task is tantamount to a declaration that it 
regards as futile the work of those who engage in it. It has been 
often said that, while seeking chimerical solutions, one may 
discover a useful truth. Such opinions might have been valid 



in days when the methods for discovering truth were equally 
unknown in all fields of endeavour; today, when these methods 
are known, it is more than probable that the surest way to find 
truth is to seek it, ..." 

This long-forgotten document reads as though it had been 
written yesterday, and not 180 years ago. What effect did it 
have? Well, if the aim of the academicians was to spare the 
scientific societies the annoyance incident to the examination 
of these solutions, then they have been more than successful. 
Soon, other academies followed suit, until today it is impossible 
for a pseudomath to get a hearing before any reputable 
scientific organization. If, however, the French Academy had 
hoped to free the world from the pseudomath, then it must be 
admitted that the document was a miserable failure. For, I 
dare say, there are more pseudomaths today than at any time 
in history; besides, their numbers increase by leaps and bounds 
from year to year, while the negative attitude of the scientific 
world, far from dampening their ardour, only makes them 
more militant. 

All this in spite of the fact that in the course of the last 
century all the problems reviewed by Condorcet were brought 
to a successful conclusion. For the pseudomath, time has stood 
still. The solutions of these problems may appear to the mathe- 
matician ever so profound and far-reaching ; to the pseudomath 
they are but mockeries, delusions and snares, to use an expres- 
sion of that king circle-squarer, James Smith. 

And, strange as it may seem, these sentiments are shared by 
the public at large. Indeed, the solutions which modern mathe- 
matics has offered to the famous problems are not solutions at all, 
as the term is usually understood. They do not culminate in a 
definite recipe prescribing certain traditional operations on 
certain traditional ingredients ; they culminate in the declara- 
tion that such a recipe is unattainable. Moreover, the reasoning 
which leads to these negative conclusions involves considera- 
tion of algebra and analysis which appear to the layman as 

F 81 


irrelevant and, therefore, wholly unconvincing. The pseudo- 
math brushes all such reasoning away with a contemptuous 
smile, branding it professional subterfuge which aims at cover- 
ing the orthodox mathematician's incompetence behind a 
smoke screen of symbols and technicalities. 


And so the merry-go-round spins on. Each year sees new solu- 
tions of the ancient problems which the benighted mathe- 
matician has long ago stricken off his list as solved. Our own 
century has been particularly prolific : dozens of solutions have 
been announced in our daily press. One of these received extra- 
ordinary publicity. It concerned the trisection problem and 
was the discovery of a president of an American Catholic 
college. Substantially, the reverend father's solution consisted 
in trebling an angle, and then exclaiming: "Behold the whole, 
and then behold the part!" Judging from the numerous 
inquiries I have received concerning it, the pater's fame must 
have travelled very far. He was, in fact, so encouraged by the 
reception accorded to his achievement that he decided to 
continue his researches, and subsequently enriched the world 
with a book in which he proved the Euclidean postulate of 
parallels and, simultaneously, annihilated the impious Einstein. 
I shall conclude this chapter by recording three conversations 
which I had, all in connection with this ecclesiastic trisection. 
My interlocutors were all college graduates. The first, a success- 
ful engineer, after listening to my explanation with ill-concealed 
irritation, interrupted me with a sneer: "The dogmatism of 
you fellows makes me tired. It reminds me of those experts who 
only twenty-odd years ago maintained that flying was impos- 
sible. Granted that the priest's construction is wrong, as were the 
other solutions before him ; what of it? To me, it only means 
that the problem is a challenge to human ingenuity. I am 
confident that some day solutions to these problems will be 
found, and, that when they are found, it will not be along the 
beaten paths which the professional mathematician is bound 
to follow." 



The second, , a literary, man with philosophical aspirations, 
said: "I cannot agree with your conclusions. It seems to me 
that mathematicians lose sight of one incontrovertible truth, 
namely, that if a problem can be formulated in certain terms, it can 
be solved in the same terms. Now, you admit that all the problems 
which you have mentioned can be formulated in terms of 
straight lines and circles; by the same token, their solutions 
should require no other lines." 

The third, a realtor, listened to my comments with com- 
placent mistrust. By way of changing the topic, or perhaps 
with a more subtle intent, he remarked that the college which 
the reverend father was heading possessed a first-class football 

Chapter Seven 


Indeed, when in the course of a mathematical invest- 
igation we encounter a problem or conjecture a 
theorem, our minds will not rest until the problem is 
exhaustively solved and the theorem rigorously proved; 
or else, until we have found the reasons which made 
success impossible and, hence, failure unavoidable. 
Thus, the proofs of the impossibility of certain solutions 
plays a predominant role in modern mathematics; 
the search for an answer to such questions has often led 
to the discovery of newer and more fruitful fields of 
endeavour. DAVID HILBERT 

One who contemplates the silhouetted skyline of a great city 
is struck with the abundance of the straight and the round to 
the practical exclusion of all other forms. In distant outline, the 
city looms as a monotony of rectilinear segments, relieved by 
an occasional arc of a circle. 

This preponderance of the straight and the round is not 
limited to the contours of the buildings where we dwell or 
work; the intricate equipment designed to aid us in our 
struggle for existence, the vehicles which transport us from 
place to place, the roads we travel, the games we play, the very 
shape of our rooms, and of the furniture, utensils and trinkets 
which crowd them bespeak this predilection. 

Even more amazing is the spectacle which awaits one behind 
the walls of our mills and shops. Round and round and to and 
fro whirl and swing the machines, lathes, drills, shapers, 
presses, ceaselessly engaged in flattening, straightening and 
turning the raw materials furnished by nature. 

Indeed, to a thinking being from another planet, unaware 


of our human purposes, the complex activity which we call 
civilization might appear as a concerted effort to force upon 
Nature, irregular in her deeds and unruly in her moods, the 
acceptance of these forms preferred by man. 

Nor is this preference an outgrowth of modern life. The 
machine age has only accentuated what has for millennia been 
latent in the human spirit. It is detected in the crude patterns 
of the savage, in the figures drawn on the walls of prehistoric 
caves. The utensils of bygone ages spared by the ravages of time 
bear mute testimony to this predilection. It is as though man 
has ever striven toward these forms as ideals, and the extent to 
which he has put them to use may be taken as indices of his 
knowledge and skill at various stages of his progress. 

Already in the naive endeavours of the primitive mind just 
awakened to the consciousness of form, in these groping efforts 
of an untutored imagination, we find, in germ, the elements 
which were destined to become the foundation of a great 
science. As time went on, these preferred forms came to be 
viewed by man not merely as indispensable principles of design 
and construction but as basic elements for an accurate descrip- 
tion of nature. To these elements he endeavoured to reduce 
the complex forms which he encountered in experience, and 
out of these endeavours grew a science which, armed and 
guided by number, eventually attained the highest levels of 

It would be fitting to call this body of knowledge the science 
of form, but, because of long historical association, it has been 
identified with one of its earliest applications, geometry, the 
measurement of earth. Under this modest name, and with such 
modest beginnings, it has gradually extended its influence over 
the physical sciences until today it bids fair to dominate any 
rational interpretation of nature. 

Yet, throughout this long evolution and to this day, the 
science has, in one respect at least, preserved its original 
character: the flat, the straight and the round, in the new guises 



of spaces, their geodesies and their curvatures, are concepts as 
basic in this cosmic geometry as they were in the rudimentary 
stages of the science. 

What has forced this choice upon man? 

One turns his back on the skyline of the great city to view 
the peaceful landscape beyond, the winding rivers, the rolling 
hillsides, the patches of marshes and forests. One contemplates 
the surrounding flora and fauna; the oddly shaped roots and 
stems and leaves and blades ; the limbs and wings and bodies of 
beast and fowl and fish and of all that creepeth upon the earth. 
No! It was not here that man has found models for the severe 
line or the smoothly rolling circle, these preferred elements of 
his manipulation and speculation. 

Why then has this distinction fallen to the lot of these special 
forms, so rarely encountered in man's natural environment? 
What is the source of this predilection, so manifest in the things 
which he has built for sustenance, comfort or defence? Why 
have these forms been chosen by man as cornerstones of that 
grandiose scheme of his own making which he seeks to identify 
with the physical universe? 

In Ancient Greece, where stood the cradle of science, the 
preference for the straight and the round took the form of an 
interdiction: the line and the circle alone could be used in 
geometric construction; all other devices, regardless of their 
effectiveness or scope, were condemned as mechanical and 
unworthy of the philosopher. 

According to Plutarch, we owe this proscription to Plato : 
"Eudoxus and Archytas had recourse to mechanical arrange- 
ments, adopting to their purpose certain curved lines and 
sections. But Plato inveighed against them with great indigna- 
tion and persistence, as destroyers and perverters of all that was 



good in geometry, which was thus lowered from the incorporeal 
and intellectual to things material, and employed besides much 
mean and vulgar labour. In this manner, mechanics was dis- 
simulated and expelled from geometry, and, being for a long 
time looked down upon by philosophers, it became one of the 
arts of war." 

Now, in appraising the historical value of these statements, 
it should be remembered that in the course of the five hundred 
years which separated Plutarch from Plato, the latter had 
become somewhat of a legendary figure whose authority was 
often invoked by contending philosophical schools in support 
of views he had never uttered during his lifetime. Thus, we 
find that other Greek historians were by no means so emphatic 
in attributing to Plato the authorship of the interdiction. 
Indeed, some go so far as to accuse the Athenian philosopher 
of having himself at one time indulged in these mechanical 
solutions so unbecoming to a geometer and gentleman. 

Whoever might have been the author of this drastic decree, 
there is ample evidence that, unlike most prohibitions, this one 
was eminently successful. In fact, it is impossible to over- 
estimate the influence which the interdiction exerted on the 
subsequent course of geometry, and, strange as it may seem, 
its effect on postclassical geometry was even greater than it 
had been during the Hellenic period. Indeed, while the pro- 
scription did succeed in drawing a very sharp demarcation 
between what we now designate as elementary geometry, 
where the line and the circle rule supreme, and the other 
branches of the science a demarcation which remains nearly 
intact today it did not prevent the Greek geometers from 
mastering the forbidden curves of which Plutarch spoke. 

Thus, as I already pointed out in a previous chapter, the 
same Euclid, whose Elements served for two thousand years as 
a model for textbooks in elementary geometry, wrote a treatise 



on the conic sections which, unfortunately, did not come down 
to us. The great Apollonius of Perga turned the study of these 
sections into a discipline as rigorous and fertile as the Euclidean 
Elements. What is more, the Greek geometers were even familiar 
with higher curves ; the curved lines of which Plutarch spoke 
are designated today as cubics, while the quadratrix of the Sophist 
Hippias is of a type called today transcendental. 

During the many centuries of decay which succeeded the 
Greek period, the achievements just mentioned had been all 
but forgotten, and when, with the revival of learning, the study 
of mathematics was again taken up, the line of demaraction 
between elementary geometry and the other branches of the 
science became sharper than ever. 

To this day we say that such and such a problem is susceptible 
of a "geometrical" solution, when all we mean is that the 
required construction can be executed by means of the two 
traditional instruments the straightedge and the compass. 
Another construction which may be effectively executed by 
means of devices as simple as either the straightedge or the 
compass we brand as impossible, only because the traditional 
instruments do not suffice here. 

This unhappy terminology contributes much to the confusion 
which the general public entertains in matters mathematical. 
The layman hears that certain problems bequeathed to us by 
antiquity are spoken of by mathematicians as impossible: he 
naturally concludes that these problems still remain unsolved. 
The fact that the last of these questions, the squaring of the 
circle, has been a closed issue for more than fifty years is rarely, 
if ever, conveyed to him. The door is thus left wide open for 
quacks to enter with their preposterous or fraudulent solutions 
of problems which demonstrably admit of no solution in the 
traditional sense of the word. 

The mathematical curricula of our schools and colleges, far 
from tending to dispel this confusion, indirectly add to it. 
However defective may be our school curricula, geography is 



not being taught out of Ptolemy, nor physics out of Aristotle; 
yet, as far as geometry is concerned, we are still in the Scholastic 
era. The textbooks used in our schools are but pale replicas of 
Euclid's Elements compiled by schoolteachers, most of whom 
are wholly unaware of the gigantic strides which geometry has 
made in the last few hundred years. As a result, the average 
layman leaves school under the impression that all that can be 
done in geometry has already been done two thousand years 
ago, with the exception of a few problems, such as the trisection 
of the angle, which still await a solution; and that here, the 
experts have admittedly failed, the mantle of glory is to fall on 
the shoulders of some amateur unpolluted by the hackneyed 
habits of the professional mathematician. 

It is a striking phenomenon, to say the least : here is a discipline 
which in modern times has so enormously increased its scope 
as to cause a veritable revolution in the scientific outlook on 
the universe, and yet, as far as its teaching is concerned, we 
might as well be in the days of ancient Alexandria. To say that 
the conservatism of our school authorities is responsible for this 
state of affairs is but to christen the difficulty. A conservatism 
so universal and so deep-seated as to withstand the onslaught 
of progress for so many centuries must have its roots in some 
inherent predilection of the human mind. 

Thus arises the question: Is there any connection between 
this deep-seated conservatism which has limited the general 
instruction in geometry to the properties of the line and the 
circle, this ancient interdiction which has proscribed the use of 
all instruments and devices other than the straightedge and the 
compass, and this inherent preference of man for the straight 
and the round so manifest in his work and his thought? 


At the risk of boring the more sophisticated reader by belabour- 
ing the obvious, I must insist that the difficulties to which the 



celebrated problems of construction lead, far from being 
inherent in the problems themselves, merely reflect the drastic 
character of the restrictions imposed on classical construction; 
that the terms possible and impossible possess no absolute signifi- 
cance ; that it is essential, in formulating any individual problem, 
to stipulate the equipment by means of which the construction is 
to be executed; that with all restrictions removed, with any 
device susceptible of mathematical definition admitted into 
geometry on equal terms with the straightedge and compass, 
and with any locus accepted on par with the line and the circle 
regardless of the mechanical or graphical procedure used in 
generating it, the terms possible or impossible lose all meaning, 
and the field of soluble problems becomes coextensive with the 
field of all problems. 

These statements are truisms, I admit. And yet, there are 
truisms which cannot be overemphasized or repeated too often. 
To this class belong those verities which stipulate the relative 
character of concepts. So intense, indeed, is man's craving for 
the absolute that his intuition is ever ready to accept arguments 
which his reason would unhesitatingly reject. The history of 
such concepts as the relativity of space and time furnishes eloquent 
evidence to this tendency of the human mind. 

In the light of these general observations, we should first of all 
examine the scopes of the traditional instruments with the view 
of ascertaining the limitations which their exclusive use imposed 
on geometrical activity. And since the restriction originated in 
Greece, we should begin by consulting Greek sources. Strangely 
enough, we find that, despite the exclusive roles which the 
straightedge and the compass played in classical geometry, 
classical treatises rarely, if ever, mentioned these instruments 
by name. In Euclid's Elements the equipment was introduced in 
the guise of postulates or common notions. The use of the straight- 
cage was sanctioned in statements that any straight line could 
be produced indefinitely, that through any two points a 
straight line could be drawn, and that two straight lines would 



merge throughout if two points on one coincided with two 
points on the other. The use of the compass was sanctioned in 
the statements that it was possible to draw a circle which had 
its centre in any point and which passed through any other 
point, and that only one such circle existed. 

As opposed to this classical tendency to keep the instruments 
in the background, the modern approach to geometrical con- 
struction puts the equipment prominently to the fore. In fact, 
the whole question could be reduced to the classification of 
problems according to the equipment they require. We could 
begin by separating problems which are susceptible of straight- 
edge-compass solutions from those which involve more intricate 
apparatus. These "higher" problems could, in turn, be grouped 
according to the character of the instruments which their 
solutions demand. For example, while the general angle may 
be trisected by a linkage, no circle can be squared by such 
means ; the latter problem may be solved, however, by means of 
a rolling mechanism, while certain other constructions necessitate 
the introduction of sliding devices. We would thus have linkage 
problems, roller problems, slide problems and many others, their 
variety limited only by human resourcefulness. 


Now, any classification scheme, no matter how cleverly 
contrived, is but an empty formality unless it is supported 
by definite criteria, i.e., by a code of unequivocal rules which 
any competent person may use to ascertain whether or not a 
given object belongs to a given class. As applied to our own 
programme, this means that we should begin by seeking 
criteria of constmctibility by straightedge and compass. It is here that 
we encounter our first difficulty. 

There is nothing, indeed, in the formulation of a construc- 
tion problem to indicate whether it can or cannot be solved by 
means of the traditional instruments. The use of ruler and 
compass enables one to trisect any rectilinear segment, but not 
the general circular arc; to inscribe into a circle a regular 
polygon of 3, 5 or 17 sides, but not one of 7, 9 or 1 1 sides; to 


square any parabolic arch, but not a circle. Such facts are not 
of the sort that may be deduced from the statement of a 
problem, nor from the casual inspection of a hypothetical 
figure; they require, as a rule, a more or less intricate and 
seemingly artificial reformulation of the problem in terms of algebra, 
and sometimes in terms of analysis. 

Is this devious approach unavoidable? The history of the 
celebrated problems gives a pragmatic answer to this query. 
For, in spite of the valiant attempts of Greek geometers, the 
problems remained at a virtual standstill for nearly two 
thousand years and were not completely solved until algebra 
and analysis had sufficiently advanced to be enlisted as effective 

As we proceed with this survey, we shall encounter many 
problems which reveal the difficulties inherent in a strictly 
geometrical approach to construction, and, incidentally, the 
reasons why the geometers of antiquity failed to resolve these 
difficulties. These problems will bring out in sharper relief the 
intimate kinship between geometrical construction, on the one 
hand, and the classification of numbers according to their 
character, on the other. In the last analysis a geometrical instru- 
ment can be identified with a category of numbers; thus, any 
restriction imposed on equipment is a restriction on number. 



Chapter Eight 


The Elements: hardly another scientific work has so 
long maintained so eminent a place in its field. Indeed, 
even today every mathematician must in one way or 
another come to terms with Euclid. 


No other proposition of geometry has exerted so much influence 
on so many branches of mathematics as has the simple quadr- 
atic formula known as the Pythagorean theorem. Indeed, much 
of the history of classical mathematics, and of modern mathe- 
matics, too, for that matter, could be written around that 

To begin with, it is the point of departure of most metric 
relations in geometry, i.e., of those properties of configurations 
which are reducible to magnitude and measure. For such 
figures as are at all amenable to study by classical methods 
are either polygons or limits of polygons; and whether the 
method be congruence, areal equivalence or similitude, it rests, in 
the last analysis, on the possibility of resolving a figure into 

Next, the Pythagorean equation being non-linear, its numerical 
applications lead to irrational numbers. In this way mathe- 
matics, almost from its inception, was confronted with the 
perplexing problem of incommensurable magnitudes, and this 
exerted a profound, even if perturbing, influence on the evolu- 
tion of the number concept. 

Again, to determine all integral solutions of the equation 

was one of the earliest problems in that branch of mathematics 



which came to be known as number theory. With the revival of 
mathematics, it led to the more general problem of determining 
integral solutions of the equation 

=.R n . . . (8-2) 

for any integral value of the exponent n. The statement that no 
such triples exist for exponents greater than 2 is known as 
Fermafs theorem. It remains in the realm of conjecture to this 

With the advent of analytic geometry, the metric aspect of 
the theorem was greatly enhanced. A direct application of the 
theorem leads to the distance formula by means of which the 
length of any segment can be calculated in terms of the coordi- 
nates of its end points. Eventually, the equation has come to be 
viewed as the analytical representative of the circles in the 
plane, and this, in turn, has led to the fertile idea of describing 
and classifying geometrical loci by means of algebraic equa- 
tions. To the same order of ideas belongs the notion of absolute 
value of a complex magnitude which plays such an important 
role in the theory of functions. 

The introduction of infinitesimal methods led to further 
extensions of the formula's scope. In the guise of a differential 
form, it became the measure of the length of the arc of a plane 
curve. The idea was eventually extended to space curves, then 
generalized to curved surfaces. 

Last but not least was the influence of the Pythagorean 
theorem on the so-called non-Euclidean geometries. When the 
axioms of geometry began to be subjected to a critical analysis, 
it was soon realized that the Pythagorean relation between 
the sides of a right triangle was equivalent to the Euclidean 
postulate of parallels. Thus, if one was to reject this postulate 
but retain the other axioms, one would have to replace the 
Pythagorean relation by another form. These considerations 
had led Riemann to the epoch-making idea of defining space 
structures by means of quadratic forms, an idea which, when 
extended to space-time manifolds, became the foundation of 
the mathematical theory of relativity. 


Let us examine the two proofs of the Pythagorean proposition 
which have been attributed to Euclid. I use the word attributed 
advisedly, because there are definite indications that the 
proof at the end of Book One of the Elements was first advanced 
by the brilliant geometer Eudoxus who antedates Euclid by 
a generation at least, while the similitude proof of Book Six 
bears the marks of the Founder, Thales. 

The characteristic feature of the first proof is that it interprets 
the Pythagorean theorem not as a metric relation between the 
sides of a right triangle but as a property of the squares erected 
on these sides. This literal interpretation of the theorem restricts 
the proof to areal equivalence. Now, to prove that noncongruent 
polygons contain the same area requires, as a rule, inter- 
mediate steps and auxiliary lines, which complicate the argu- 
ment and obscure the figure. This may explain why the proof 
of Book One has been a source of despair to so many beginners, 
and why even those who have grasped it can rarely throw off 
the feeling that the proof is artificial and unnecessarily intri- 
cate. Thus, the caustic German philosopher Schopenhauer 
dismissed the demonstration with the contemptuous remark 
that it was not an argument but a ' 'mousetrap." 

Many a textbook on geometry has been written in the 
twenty-two-hundred-odd years since Euclid's work appeared. 
Some are mere facsimiles, others but blind adaptations of the 
Elements. Still, there are quite a few among these that make 
some pretence to originality. But even the latter present the 
"mousetrap" as the proof of the Pythagorean theorem, while 
the elegant demonstration of Book Six is rarely, if ever, 
mentioned. And yet, not only is this alternate proof superbly 
simple, but, by identifying the Pythagorean relation with 
the existence of similar figures, it strikes at the very root of the 
question. Paraphrased in modern terms, this means that in a 
geometrical field where two figures cannot be similar without 
being congruent at the same time, the relation between the 
sides of a right triangle would not be of the Pythagorean form. 

o 97 


The similitude proof, Proposition 31 of Book Six of the Elements, 
derives from a property which is characteristic of right 
triangles. The principle is illustrated in Figure n. The perpen- 
dicular dropped from the vertex C of the right angle onto the 





H u- 











C LI' 


hypotenuse partitions ABC into the two right triangles AHC 
and BHC; either of these is equiangular with the original triangle 
and, therefore, similar to it. Hence the two proportions, 

u : a = a : c y and v : b = b : c. 
From these we draw 

and, by addition, 

= cu, and i 2 = cv, 

a* + b 2 = c(u + v) = c 2 . 


Figure 1 1 gives an interpretation of these relations in terms of 
areas. The extended altitude of the triangle ABC partitions the 
square erected on the hypotenuse into the two rectangles 
(HA') and (HB')\ in virtue of the preceding relations, the 
areas of these rectangles are a 2 and # 2 , respectively. This, as 
we shall presently see, is the property which Euclid in his 
earlier proof sought to establish, but without the benefit of 
similar triangles. 


As a matter of record, to Euclid the proposition of Book Six 
was a generalization of the Pythagorean theorem rather than 
an alternate proof. Here is his wording: "In right-angled 
triangles, the rectilinear figure erected on the hypotenuse is 
equal to the similar and similarly described figures upon the 
sides containing the right angle." The drawing in the Elements 
shows three similar rectangles (Figure 12). However, the 
"described figures" could be equilateral triangles, or, for that 
matter, three semicircles. 



The key to the areal proof of the Pythagorean theorem, Proposi- 
tion 47 of Book One of the Elements, is a lemma which permits 
one to transform any given triangle into another equal to it 
in area but not necessarily congruent to it. The lemma is 
exhibited in Figure 13, where MN is a fixed segment, xx an 
indefinite line parallel to Af JV, and X any point on the line xx. 
As the point X moves along the line, the triangle MXN is 
deformed, but its area remains unaltered. 


M'X S x 4 x s 

x 7 




Once this lemma is borne in mind, the stratagem of the 
Euclidean proof becomes clear. The aim is to establish that the 
triangles HBQ, and MEN in Figure 14 are equiareal. In virtue 
of the preceding lemma, HBQ, can be replaced by the equi- 
valent triangle CBQ, and MNB by the equivalent triangle 
ANB. However, in the triangles CBQand ANB we have: 

AB = QB, CB = BN, and angle QBC = angle ABN\ 

thus CBQ, and ANB are congruent and, therefore, equiareaL 
Passing from triangles to rectangles, one finds BHKQ, equal 
to BCMN, and, by analogous reasoning, AHKP equal to 




Why did Euclid place the intricate areal proof in the fore- 
ground, relegating the simple similitude proof to the very end 
of his study on similar figures? The answer is that the author 
of the Elements^ with his customary thoroughness, would not 
deal with relations which depended on the principle of similitude 
until he had made an exhaustive study of ratios and proportions. 
Book Five of the Elements is just such a study. 



One who reads the Elements today is amazed at the pains- 
taking care and tedious detail with which Euclid treats certain 
notions, notions which modern curricula dismiss with a few 
casual remarks. Particularly perplexing is the bewildering 
variety of special cases handled, cases which we would regard 
as but trivial variations of a general rule. One explanation of 
this apparent verbosity is to be found in the rudimentary state 
of Greek algebra. In the absence of an adequate symbolism, 
Greek mathematicians resorted to verbal procedures; these 
were eventually codified into a glossary of terms and rules 
most of which strikes us today as utterly superfluous and hence 


A case in point is the theory of proportions which constituted a 
very important part of the rhetorical algebra used by Greek 
geometers to express metric relations. It is here that the discrep- 
ancy between classical and modern exposition appears most 
striking. To illustrate, consider the proportion, a : b = a' : b f , 
and its "counterpart," the equality, ab r a'b. Observe the 
facility with which we pass from one of these expressions to 
the other, then reflect that what we view today as a trivial 
manipulation of algebra was to the Greeks a basic theorem of 

The theorem assigns to any pair of similar rectangles another 
pair of rectangles, equal in area. Associated with this proposition 
is a configuration which is shown in Figure 15. Let be any 
point inside the parallelogram ABCD: the lines through 
drawn parallel to the sides of the parallelogram partition the 

1 02 


latter into four equiangular parallelograms (OA) 9 (05), (0(7) 
and (OD) which I shall call cells for short. The theorem in 
question is this: If the cells (OA) and (OC) are similar, then the 
cells (OB) and (OD) are equiareal, and ^conversely. The similarity 
condition is fulfilled if, and only if, the point O lies on one or 
the other of the diagonals of ABCD. One thus arrives at the 
geometrical counterpart of the relation: 

a : b = a' : b' entails ab' = a'b, 
and conversely. 

It would be a mistake, however, to ascribe Euclid's meticulous 
handling of proportions solely to the low state of Greek algebra. 
The truth is that the Greeks were reluctant to identify ratios 
with numbers, the compelling cause of this hesitancy being the 
existence of incommensurable magnitudes. 

How could one define the ratio of a circumference of a circle 
to its diameter, or the ratio of the diagonal of a square to its 
side, for that matter, without resorting to infinite processes and 
their limits, with all the qualms that such notions are heir to? 
Euclid was a Platonist, and the echoes of the controversy 
engendered by the Zenonian paradoxes had not yet subsided 
in his time. And lest we be tempted to treat this caution with 
too much levity, let us recall the travails of the modern theory 
of irrationals, the critical studies of Weierstrass, Cantor, Dede- 
kind and Poincare, the antinomies and the bitter controversies 
which are still fresh in the memories of the older mathe- 
maticians of our generation. 

Figure 16 is a schematic drawing of what may be called an 
"articulated" ruler. The longest link AB is 5 units long; the 
shorter links measure 3 and 4 units, respectively. If the outer 
links be revolved about the pivots B and C until their end 






points meet in A, the resulting triangle ABC would have a 
right angle at C, in virtue of the identity, 

3 2 + 4 2 = 5 2 - 

This method of constructing right angles antedates Pytha- 
goras by thousands of years. It was used by ancient Egyptian 
surveyors in orientation problems; by the Chinese as a levelling 
device in masonry work; in other Oriental lands as sort of 
carpenter square. There is ample evidence that the ancients were 
aware of the existence of other rational triangles, such as are 
given by the triples (5, 12; 13) and (7, 24; 25), and it was, 
undoubtedly, the search of these triples that had led the 
early Greek mathematicians to the Pythagorean theorem. 

The latter was a triumphant confirmation of the Pythagorean 
number philosophy. However, the triumph was short-lived, 
for, the very generality of the proposition revealed the existence 
of irrational magnitudes. One effect of this perturbing discovery 
was a revised outlook on matters of geometry. To the early 
Pythagoreans every triangle was a rational triangle, because they 
held that all things measurable were commensurate. This last dictum 
seemed to them as incontrovertible as any axiom, and when 
they proclaimed that number ruled the universe, they meant by 
number integer, since the very conception that magnitudes 
might exist which were not directly amenable to integers 
was alien to their outlook as well as to their experience. 



Some modern interpreters of mathematical thought have been 
inclined to dismiss such ideas as naive notions of a bygone 
age. And yet in the eyes of the individual who uses mathe- 
matical tools in his daily work and his name today is legion 
but to whom mathematics is but a means to an end and never 
an end in itself, these notions are neither obsolete nor naive. 
For, such numbers as are of practical significance to him 
result either from counting or from measuring and are, therefore, 
either integers or rational fractions. To be sure, he may have 
learned to use with comparative facility symbols and terms 
which allude to the existence of nonrational entities, but this 
phraseology is to him but a useful turn of speech. In the end 
the rational number emerges as the only magnitude that can 
be put to practical use. 

Should this individual, piqued by the reproach that he was 
naive, endeavour to penetrate behind the mysterious nomen- 
clature, he would soon discover that the processes invoked to 
vindicate these nonrational beings are wholly unattainable 
and, therefore, to him gratuitous. And yet, should he persist in 
his attempt to interpret such an entity in his own rational 
terms, he would be sternly reminded that in matters irrational 
one may at times evade the infinite but never avoid it. For, 
inherent in the very nature of this imponderable magnitude is 
the property that no matter how close any given rational 
number may "resemble" it, other rational numbers exist 
which "resemble" it even closer. 

This individual would feel far more at home among the 
Pythagoreans than among their rigorous successors. He would 
willingly embrace their credo that all things measurable are 
commensurate. Indeed, he would be at a loss to understand 
why a principle so beautiful in its simplicity has been so 
wantonly discarded. And, in the end, the mathematician 
would be forced to concede that the principle was abandoned 
not because it contradicted experience, but because it was 
found to be incompatible with the axioms of geometry. 



Indeed, if the axioms of geometry are valid, then the Pytha- 
gorean theorem holds without exceptions. And if the theorem holds, 
then the square erected on the diagonal of a square of side i 
is equal to 2. If, on the other hand, the Pythagorean dictum 
held, then 2 would be the square of some rational number, and 
this contradicts the tenets of rational arithmetic. Why? 
Because these tenets imply, among other things, that any 
fraction can be reduced to lowest terms; that at least one of the 
terms of such a reduced fraction is odd; that the square of an 
odd integer is odd ; and that the square of an even integer is 
divisible by. 4. Suppose then that there existed two integers, 
x and R, such that R 2 /x 2 = 2, i.e., that 

It would follow that R was even, and, since the fraction is in 
its lowest terms, that x was odd. One would thus be led to the 
untenable conclusion that the left side of an equality was 
divisible by 4, while the right was not. 

The preceding argument is a modernized version of Euclid's 
proof of the irrational character of ^ 2. Its very simplicity hints 
that it was but an adaptation of an earlier proof, perhaps, of 
the very one that had so profoundly perturbed the Pythagoreans 
and eventually forced them to change their outlook on number 
and measure. 

Obviously, the reasoning is not limited to the case of an 
isosceles right triangle. Consider any right triangle the sides of 
which are integers, say x andjy, but such that x 2 +JV 2 is not a 
perfect square: by an argument patterned on that of Euclid it 
can be readily proved that such a triangle is not rational. 
This clearly reveals the existence of an infinitude of non- 
rational triangles. What is more, the preponderant majority of 
construction problems of classical geometry depended on just 
such nonrational triangles: it is sufficient to mention the 
golden section., the regular polygons and bisection of standard angles. 

As a matter of fact, the determination of rational triangles 

1 06 


is not a problem of geometry, since geometrically there is no 
way of distinguishing a rational triangle from any other. 
Nor is it a problem which algebra could resolve, inasmuch as 
the laws of formal algebra are obeyed by irrational numbers 
as well as rational. In the last analysis the problem is a study 
in integers. 


Chapter Nine 


We found a beautiful and most general proposition, 
namely, that every integer is either a square, or the 
sum of two, three or at most four squares. This 
theorem depends on some of the most recondite 
mysteries of numbers, and it is not possible to present 
its proof on the margin of this page. 


This chapter will deal with integers, more particularly 
with positive integers, i.e., with the "natural" sequence, 
i, 2, 3, 4, . . . , the starting point of all mathematics. The 
branch of mathematics dedicated to the study of integers has 
come to be known as theory of numbers. The origin of this rather 
unhappy terminology is obscure, but one thing is certain : the 
misnomer cannot be blamed on the Greeks. 

Classical Greek had two distinct words for number : arithmos 
for integer, logos for general number, and although in lay 
literature the two terms were at times loosely used, the mathe- 
matical writers were fairly consistent in distinguishing between 
their meanings. Thus the theory of numbers was called arith- 
metica, while what we today call arithmetic was then logistica. 
This last word, which has survived as a military term, may be 
traced to the Greek logisticos, calculator, more particularly a 
calculator attached to an army, charged with the planning, 
figuring and procuring supplies and equipment. 

The misnomer harks back to an age when number meant 
positive integer , and little else. We have travelled a long way 
since; the evolution of the number concept has become one 
of the most profound, fertile and far-reaching mathematical 
studies, and it would be fitting indeed to call this larger study 
theory of numbers. So it is rather regrettable that the title has 

1 08 


been pre-empted by a study which deals with integers exclu- 
sively. It may be argued that once this glaring misnomer has 
been recognized for what it is, it could be readily corrected, 
but one who harbours such thoughts underestimates the power 
of tradition, even in matters mathematical. 

It is true that any systematic study of the general number 
concept must take the integer for point of departure, and so 
it may seem that the theory of integers is a sort of introduction 
to the science of number, and that as such it deserves the name 
bestowed upon it. We find, however, on closer scrutiny, that 
the chief preoccupation of the theory of numbers, as it is 
cultivated today, is not with integers at large but with particular 
types of integers, studied either individually or in sets. So 
special, indeed, are some of the problems of the theory that 
the keenest tools of analysis are often not keen enough to 
pierce the difficulty, with the result that quite a few of these 
problems remain unsolved to this day. 

The challenge of these problems spurred the efforts of the 
greatest mathematicians from Fermat to Hilbert. Special 
methods of utmost ingenuity have been devised by these 
masters, and these methods have, in turn, enriched other 
branches of mathematics. With all that, the theory of numbers 
remains a sort of sui generis of mathematics, the magnificent 
pinnacle of an edifice rather than an integrated part of its 
structure. Like virtue, number theory is its own reward. 
Indeed, it is altogether possible to carry on studies in practically 
any branch of modern mathematics without ever facing the 
necessity of using number- theoretical tools. In this our age of 
extreme specialization, most mathematicians may safely remain 
ignorant of number theory, and most of them do. Gauss pro- 
claimed the theory Regina Mathematica, but to the modern 
mathematician the queen is largely a figurehead. 

The so-called Pythagorean problem may be stated as follows: 
to determine all integer sets which satisfy the equation 



I call such sets Pythagorean triples, or triples for short. The 
terms x and y are the sides of the triple, R its hypotenuse. Any 
triangle the sides of which can be represented by integers will 
be called a rational triangle. This does not restrict the scope of 
the term "rational" for if a triangle can be represented by 
means of rational numbers , i.e., integers and fractions, then it 
can also, through a proper change of unit, be represented by 
integers alone. 

The branch of number theory which deals with integral 
solutions of equations is called Diophantine analysis, named after 
the Alexandrian mathematician of the fourth century A.D. 
who, as far as we know, was the first to attack such problems 
in a systematic manner. 

The Diophantine equation (9.1) is of a special type called 
homogeneous. The important feature of such equations is this : 
if (x, y ; K) is a set of values which satisfy a homogeneous equa- 
tion, then the proportional set (nx, ny ; nR) is also a solution of 
the equation, whatever value be assigned to n. Because of the 
homogeneous character of the Pythagorean relation, we can 
classify triples into primitive and imprimitive. A triple is primitive 
if its terms are relatively prime, i.e., have no divisors in common. 
On the contrary, the terms of an imprimitive triple have some 
divisors in common. Examples of primitive triples are (3, 4; 5), 
(5, 12; 13), (8, 15; 17); examples of imprimitive : (9, 12; 15), 
(10, 24; 26), (80, 150; 170). 

Associated with every primitive triple (x,y\ R) is an infinitude 
of imprimitive: (2*, 2?; zR), (3*, <$y, 3^), (4*, 47; 4^), . 
(nx, ny, nR), .... On the other hand, if (x,y, R) is an imprimi- 
tive triple, it is always possible to determine a primitive triple 
the terms of which are proportional to x, y and R; and this by 
the simple expedient of dividing every term of the triple by 
its greatest common divisor. I shall call this operation contraction. 

Thus, whether any given triple is primitive or imprimitive 
depends on the value of its greatest common divisor. The labour 
incident to calculating this divisor is greatly facilitated by the 
following theorem, which is a direct consequence of the homo- 
geneous form of the Pythagorean relation: any integer which 
divides two terms of a Pythagorean triple also divides the third term. 

This theorem has two practical corollaries:/^, to determine 



the greatest common divisor of a triple it is sufficient to calcu- 
late the greatest common divisor of any two terms of the 
triple ; and second, if any two terms of a triple are relatively prime 
then the triple is primitive. As an example, consider the set 
(140, 171; 221). Is it a triple, and if so, is the triple primitive? 
The set is a triple, because : 

22i 2 iyi 2 = (221 + iyi)(22i 171) 

= 392-50 = 196-100 = no 2 ; 

and the triple is primitive, because 140 and 171 are coprime. 
On the other hand, the set (36, 105; in) is not primitive; its 
greatest common divisor is 3, and the contracted primitive triple 
is(i2, 35; 37). 

A comprehensive solution of the Pythagorean equation was 
reserved for modern times. Though there are many allusions 
to the question in Book Ten of Euclid's Elements, in the Arith- 
metica of Diophantus and in several other mathematical tracts 
of the classical period, no systematic approach to the problem 
was even attempted. One reason for this was already brought 
out: considerations of rationality played a rather minor role 
in classical geometry, and geometry was the paramount 
interest of Greek mathematicians. 

This much can be affirmed with certainty. The Greeks 
were fully aware of the importance of the concept ofprimitivity; 
they knew that one of the sides of a primitive triple was even, the 
other odd, and that the hypotenuse was always odd', they knew how 
to generate certain types of triples in number indefinite, and 
concluded from this that the aggregate of primitive triples was 
infinite. I shall elaborate somewhat on their arguments and 
methods, in modernized version, of course. 

Let us take up the question of parity first. The case of three 
odd terms is out, on the ground that the sum of two odd integers 
is even; while the case of more than one even term is eliminated 
on the grounds of primitivity. Thus, there is only one even 
term, and it remains to show that it is not the hypotenuse. 



Assume the contrary, and set x = 2 + i, y 20 + i, 
j? = 2W. Substituting, we obtain 

which is untenable, since the right member is divisible by 4, 
while the left member is not. 

Now, it has been conjectured that the Pythagoreans knew 
that any odd integer may serve as one side of a primitive triple, and 
that they discovered this property while seeking to determine 
triples two terms of which were consecutive integers. It is a 
plausible conjecture, because the proof of the theorem is quite 
within the ken of the period. Let us assume that y and R are 
the consecutive terms, and set y = 2p, R = 2p + i. Then, 
substituting and solving the Pythagorean equation for x, we 
obtain x = ^ \p + i. Now, any odd square is of the form 4p + i, 
and consequently we can choose for x any odd integer. As an 
example, let us set x = 9: then x 2 = 81, and consequently 
p = 20; thus, y = 40 and R = 41. Generally, by setting 
x = 2s + i , we obtain the triple 

X = 25 + I, y = 2S(S + i), /Z = 2S 2 + 2J + I. . (9.2) 

These triples are primitive for all values of the parameter s, 
since two consecutive integers are always relatively prime. By varying 
s from i to oo, we obtain an infinite aggregate of distinct primitive 

The Platonists, on the other hand, were more concerned 
with generating triples two terms of which were consecutive 
odd integers. Proceeding as before, we set 

x 2p i, R = 2p + i, 

and derive y 2 *J 2p- ^ follows that 2p must be an even 
square. By putting zp = 4^2, we obtain the triple: 

x =4s 2 - i,y =4*, R =4* 2 + i; . (9.3) 

and the set is primitive for any value of the parameter s, because 
two consecutive odd integers are always coprime. Thus there is a 
triple associated with every term of the progression 

4, 8, 12, 16, . . . , 4s ..... 



Fibonacci sought to extend the preceding results by determin- 
ing primitive triples, given the difference between the hypo- 
tenuse and the even side, and discovered that the problem 
had no solution unless the stipulated difference was itself a perfect 
square. This led him to the idea of representing Pythagorean 
triples by means of two parameters, an idea which was a turning 
point in the history of the problem. 

The official name of this gifted Italian mathematician of the 
early thirteenth century (perhaps the only European of the 
Middle Ages worthy of the title) was Leonardo of Pisa. His 
father was a lowly shipping clerk nicknamed Bonaccio, which, 
in the idiom of the period, meant simpleton. Hence, Fibonacci, 
son of a simpleton. Nor was this the only compliment paid to 
Leonardo by his fellow citizens : he was also called Bigollone, 
i.e., blockhead. In proud defiance of these indignities, Leonardo 
adopted both nicknames as pen names. The title of the book 
in which the ideas just mentioned were first introduced may 
sound quite glamorous in Latin, but an unvarnished translation 
would read : A Book on Quadratics, Written by Leonardo the Block- 
head, Son of the Simpleton of Pisa. 

Fibonacci was the first to combine Greek achievements in 
geometry and number theory with the algebra of the Arabs 
and the positional numeration of the Hindus. In fact, his 
earlier work, Liber Abacus, should be viewed as an attempt at 
vindicating the principle of position. The book extols the many 
advantages of the new method over the traditional Roman 
numeration which was still widely used at the time; this may 
account for the abundance of examples drawn from the 
flourishing commercial life of the period. He also wrote a 
book on geometry, and while he added little to the store 
bequeathed by the Greeks, he was a pioneer in the applications 
of algebra to problems of geometry. 

But his best contributions belong to number theory. It was 
he who conceived the idea of generating arithmetical sequences 
by means of algorithms. He knew many of the identities which 
we associate today with the names of Vieta, Euler or Lagrange, 

H 113 


and made skilful use of these. It is true that his reasoning was 
occasionally tinged with error; still his demonstrations were, on 
the whole, remarkably rigorous for his period. 

Fibonacci's approach to the Pythagorean problem is sub- 
stantially this : let y be the even side of a primitive triple of 
hypotenuse R\ then the integers, R -\- y and R y are relatively 
prime , for, if they had a common divisor, say Z), then D would 
divide their sum and their difference, i.e., zR and 27, and this 
is impossible, since R andjv are assumed coprime, and R +y 
and R y are odd. 

We next invoke a lemma which, in spite of its utmost 
simplicity, plays a capital role in many number-theoretical 
arguments: let ^4, B } C, . . . be any number of positive integers 
with no divisors in common] if 

ABC, ...=,. . . (9.4) 

i.e., if the product of these integers is a perfect n th power, then each of 
the factors is an n th power "in its own right" and we may infer that 

A = a n , B == b n , C = c n , . . . . (9-4') 

where a, b, c, . . . are also relatively prime integers. 
As applied to the Pythagorean equation, we have 

and we conclude of the existence of two integers u and v such 

R+y=u* 9 R-y=v*. . . (9.5) 

Expressed in terms of these integers, the sides of the triple 

x = uv, 2y = u 2 v 2 , zR = u 2 + v*. . (9.6) 

Since u and v are odd, u + v and u v are even. This suggests 
the substitution u + v 2p, u v = 2*7, which puts (9 . 6) in 
the more convenient form 

* =/> 2 - ? 2 , y = *pq, R=p+q*. . (9.7) 


A "random" choice of the integers p and q would yield a 
Pythagorean triple, but the triple would not be primitive, 
unless the parameters p and q were relatively prime and of opposite 
parity. Indeed, if p and q had a common divisor, say Z>, then 
D would also divide x,y and R; and if p and q were both odd, 
then p 2 q 2 and p 2 + q 2 would be even. Thus, a necessary 
condition for the primitivity of the triple represented by equations 
(9 7) ^ thrt one f th e parameters be odd, the other even, and that 
the parameters be relatively prime. These conditions are also 

The real significance of the Fibonacci approach is that it 
reduces the Pythagorean problem to "two degrees of freedom," 
and does it in an "exhaustive" manner. By this I mean that 
any primitive solution of the equation 

=R 2 

can be expressed in terms of two integers p and <?, the latter 
being coprime and of opposite parity. 
The relation 

R=p* + q* . . . (9.8) 

suggests a systematic method of generating Pythagorean 
triples, as shown in the table on the following page : any entry is 
the sum of an even and an odd square and is, therefore, the hypo- 
tenuse of some triple. The "blanks" result from adding squares 
which are not coprime, and, therefore, correspond to irn- 
primitive solutions. 

To what extent does the knowledge of one term of a primitive 
triple determine the remaining two? The fundamental formulae 
of the preceding section answer this question as follows : 

First: Any odd integer is a side of at least one primitive triple. 
Indeed, any odd integer may be written in the form 

x==ia l)fcY... = A-B-C . . . (9.9) 



Hypotenuses of Primitive Triples: R = p 2 + 

Even Squares 




64 100 

144 ... 

r i 




65 101 

145 ... 




73 109 






I6 9 ... 






"3 H9 






145 181 

. , . 




185 221 

265 . . . 

where A, B, (7, . . . are odd integers, relatively prime in pairs. 
There are, generally, several ways in which x may be resolved 
into a product of two relatively prime integers, say M and N. 
For each one of these combinations, we may set p + q = M 
and p q = jV, and obtain a distinct triple. One particular 
representation is x = x i and this leads to the triples con- 
sidered in Section 3, in which the even side and the hypotenuse 
are consecutive integers. 

Second: If y is an even integer, but not a multiple of 4, then no 
solution of the Pythagorean equation exists; on the other hand, any 
multiple of 4 is a side of at least two primitive triples. Indeed, any 
multiple of 4 may be written in the form 


y = 

where k > 2, and A 9 B 9 C 9 . . . are odd and relatively prime 
in pairs. Proceeding as before, we find that there are, generally, 
a variety of ways to present \y as the product of two coprime 
factors of which one is even and the other odd. Among these 
there is always the choice p = \y, q = i, and this leads to the 



Platonist type discussed in Section 3, in which two terms 
are consecutive odd numbers; but there is also the choice: 
^ =2 *-i, q=A-B*C, .... 

Third: Under what conditions will a given integer R be the hypo- 
tenuse of some primitive triple? One may say that R must be the 
sum of two relatively prime squares of opposite parity, but 
that, in a sense, is just begging the question. How, indeed, is 
one to ascertain whether a given odd integer R is or is not 
representable as a sum of two squares, particularly if R is 


A partial answer to this question was a theorem which Fermat 
stated without proof in a letter to Father Mersenne, dated 
1640. The proof was given by Euler in 1754, and later simplified 
and extended by Lagrange, Legendre and Gauss. It eventually 
led to far-reaching investigations into the arithmetic properties 
of quadratic forms . These, however, do not concern us here, and 
so I shall confine myself to a statement of Fermat's theorem 
and to an outline of its immediate applications to the Pytha- 
gorean problem. 

Let us observe, first of all, that any odd number is either of 
the type 472 + i or of the type 472 + 3, which is but another 
way of saying that if an odd number is divided by 4, the 
remainder is either i or 3. In the second place, an odd square 
is necessarily of the type 4/2 + i , because 

(*p + i) 2 = tf(p + i) + i = 4* + I- 

Next, let us agree to designate as 2-square any integer which 
can be represented as the sum of two squares: then an odd 
2-square is always of type 4n + i, since its components, p 2 and 
q 2 , are of opposite parity. It follows that any admissible hypo- 
tenuse of a primitive triple is of type 4/z + i> i-e., a term of 
the arithmetic progression 

5> 9> 13> j7> 21, 25, 29, 33, 37, 4> 45> 49> 53, 57, 61, 65 , ---- 

. (9- 1 1) 


The underlined numbers are 2-squares: observe that these 
integers are either prime, such as 5, 13, 17, 29, . . . , or products 
of prime 2-squares, such as 65, 85, 117, . . . , or powers of prime 
2-squares, such as 25, 125, 169, .... It must have been such 
an empirical study that had led Fermat to his discovery. 

Fermat's 2-squares theorem may be stated as follows: any 
prime number of type 4n + l can oe partitioned into a sum of two 
squares, and the partition is unique. Conversely, if the equation 

has only one solution in x and y, then R is a prime. Further- 
more, if the equation (9.12) has any solutions at all, then R 
is either a prime 2-square or a product of prime 2-squares. The 
bearing of these propositions on the Pythagorean problem is 
this : a necessary condition for an odd integer R to be the hypotenuse 
of a primitive triple is that every one of the prime divisors of R be of 
type 4n + i. This condition is also sufficient. 

Thus the hypotenuse of a primitive triple is not divisible 
by 3, 7, 1 1, 19, or by any prime of type 4^ + 3- This restriction 
does not apply to the sides, x andjy, of the triple. Indeed, one 
or the other of the sides must be a multiple of 3. This is a 
direct consequence of equations (9.7), according to which : 

-q*). . . . (9.13) 

For, if p is not a multiple of 3, then p* is of type 37* + i ; if 
neither p nor q is a multiple of 3, then both p 2 and g 2 are of 
type 3/2+1, and, consequently, their difference is divisible 
by 3. Thus xy is always divisible by 3, and, as a matter of 
fact, by 12. 

On the other hand, the hypotenuse R may be divisible by 5 ; 
what is more, one of the three terms of a primitive triple 
must be divisible by 5. Here is a simple "digital" proof of this: 
the fourth power of any integer/? ends in o or 5, if/? is divisible 
by 5; and ends in i or 6, if p is not. It follows that if neither 
p nor q is a multiple of 5 then p* q* is divisible by 5. Thus if 



the even side y is not a multiple of 5, then the product Rx 
is divisible by 5. As a combination of these properties we infer 
that the product of the three terms of a primitive triple is always 
divisible by 60, a fact which was known to Fibonacci. 


In its formal aspects, the Fermat theorem is but a paraphrase 
of an algebraic identity which, in "rhetorical" form, was 
already used by Fibonacci, namely: 

(P 2 + q*)(P'* + <?' 2 ) = (PP' + qq'} z + (pq 1 -P'qV 

= (PP' - qq'Y + (pq' + P'q}*- (9-14) 

In the special case when p' = p and q' = q, the second part 
reduces to 

The last was used by Fibonacci to establish the compatibility 
of Equations (9.7). Yet, for all their importance, these iden- 
tities are purely formal, by which I mean that they apply not 
only to integers but to any entities which obey the laws of formal 

These identities show that the product of any number of 
odd 2 -square integers is itself a 2 -square, and, therefore, of 
type 4^ + i . On the other hand, not every integer of this type 
is a 2-square. The crux of the difficulty lies in the circumstance 
that the product of two integers of type 4/2 + 8 is of type 
4?z + i . For an odd integer R to be the sum of two squares it 
must be of the form 4^+1; but this is not sufficient. However, 
it is sufficient in the case when R is a prime number, and this 
is Fermat's theorem. 


Viewing the problem in retrospect, can one claim that after 
engaging the efforts of first-rate mathematicians for twenty-five 
hundred years the Pythagorean equation has finally been 


exhaustively solved? The answer will depend on what one is 
willing to accept as solution. An odd integer R of type 4/2+1 
being given, the problem is to ascertain whether a primitive 
triple of hypotenuse R exists, and if it does exist, to determine 
the sides of the triple. Fermat's theorems reduce the question 
to determining the prime divisors of R, and this may at first 
sound like a solution of the problem. Unfortunately, it is one 
of those cases which were so aptly described by Eratosthenes 
as "replacing one perplexity by another even more per- 
plexing." Indeed, not only do we lack effective criteria for 
testing the primality of integers, but the available practical 
means are so limited as to render the task formidable beyond 
imagination, when the integer exceeds 1,000,000. 

To illustrate this last point, let us consider the integer 
1,000,009. In a paper published in the year 1774, Euler listed 
this integer as prime. In a subsequent paper Euler corrected 
his error and gave the prime divisors of the integer, adding 
that at one time he had been under the impression that the 
integer in question admitted of the unique partition 

(a) 1,000,009 i,ooo 2 + 3 2 

but that he had since discovered a second partition, namely, 

(b) 1,000,009 = 235 2 + 972 2 , 

which revealed the composite character of the number. 

Euler then proceeded to calculate the divisors of 1,000,009 
by a method patterned along the proof of a Fermat theorem 
which he gave in an earlier paper, and which stated that if 
an odd integer R is susceptible of more than one partition into two 
squares, then R is composite. The interesting thing about this 
method is that it not only proves the existence of the divisors 
but permits one to calculate them in terms of the elements of 
the given partitions. Thus in the case under consideration, 
Euler found 

(c) 1,000,009 == 293 X 3,413. 

Since both divisors are prime numbers, no third partition 


Chapter Ten 


Geometry may at times appear to take the lead 
over analysis, but in fact precedes it only as a servant 
who goes before his master to clear the path and 
light him on the way. 


He flourished in the middle of the fifth century B.C. as did that 
other Hippocrates, whose pledge healers honour, even in the 
breach. I doubt that the two had ever met or even heard of 
one another. Indeed, the name was common among ancient 
Greeks who, seemingly, held horse and horseman in much 
esteem; hence, "hippo" in prefix, as in Hippocrates, Hippias, 
Hipparchos; or in suffix, as in Philippos, Speusippos, Xant- 

Hippocrates, the geometer, haled from Chios, an island 
which lies a hundred-odd miles from Miletus, the birthplace 
of Thales. Indeed, the legend of his life resembles that of the 
Founder in more than one way. He, too, began as a merchant 
and ended up as a teacher; he, too, was initiated into the 
mysteries of number and extension after reaching maturity. 
However, his mentors were not priests but zealous followers 
of the Pythagorean doctrine which by that time had grown 
into a veritable cult. For, we are told that while on a visit to 
Athens he came in contact with a group of Pythagoreans who 
taught him what they knew of geometry and arithmetic ; that 
having subsequently lost his fortune, he was reduced to selling 
these mathematical secrets to anyone who could and would 
pay the price, thus betraying his mentors' trust; that this 
sordid traffic had roused the righteous wrath of his erstwhile 
teachers who henceforth countered his achievements with 
contemptuous silence, 



And how did he lose his fortune? Well, one version was 
that his ships had been plundered by pirates on the high seas; 
but Aristotle, who never missed a chance to vent his spleen 
against mathematicians, gave a less glamorous account of the 
event. "It is well known," he wrote, "that persons brilliant in 
one particular field may be quite stupid in most other respects. 
Thus Hippocrates, though skilled in geometry, was so supine 
and stupid that he let a customs collector of Byzantium swindle 
him out of a fortune." 

The scope of his contributions is a moot question, for, while 
the Pythagoreans ignored his work, their opponents swung to 
the other extreme. He, allegedly, wrote a treatise on geometry, 
the first of its kind, where among many other innovations he 
introduced the use of capital letters to designate points on a 
figure. No trace of the textbook remains, but the technique 
of describing a figure by means of letters placed at salient 
points has since become universal. 

He is credited by Eratosthenes with reducing the Delian 
riddle to the insertion of two mean proportionals between a segment 
and its double, thus paving the way to all subsequent solutions 
of the problem. Some assert that he was the first to prove that 
the area of a circle was equal to that of a triangle erected on 
the semicircumference as base and radius as altitude, thus 
reducing the problem of squaring a circle to the rectification of 
its boundary. Others go so far as to claim that he was the 
first to advance the epoch-making idea of viewing the area 
within a closed curve as the limit of a variable polygon inscribed 
in the boundary. 

How much of this is fact and how much fancy we shall 
probably never know. So let us pass to the one achievement 
which both friend and foe associate with the name of the 
man : the Hippocratean crescents. 

Broadly speaking, a crescent or lune is a portion of the plane 
bounded by two circular arcs. However, in what follows we 
shall be concerned only with the case when both arcs lie on 



one side of their common chord, CD in Figure 1 7. The axis 
of symmetry of the crescent, or meniscos, as the Greeks called 
it, contains the centres A and B of the circles, as well as the 
midpoints E and F of the arcs. In fact, the crescent is com- 
pletely determined by the triangle ABC; and so the first step 
in squaring the crescent should be to express the area in terms of 
the elements of that triangle. Yet, Hippocrates did nothing of 
the sort; for that matter, he couldn't if he would, and wouldn't 
if he could. 


In the first place, the connection between the area of the 
crescent and the elements of the triangle involved concepts 
which lay beyond the scope of classical geometry. Indeed, 
denote by F the area of the crescent, by A the area of the 
quadrilateral ACBD y by a and b the radii of the circles, and 
by 2 a and 2/? the central angles subtended by the two arcs; 
then, a direct examination of the figure leads to the relation : 

F + Sector CADEC = A + Sector CBDFC. 


From this we draw the formula 

T = A + (0 2 /? - b*a) = ab sin (/? - a) + (a 2 /? 

. (10. i) 

a formula which involves the angles a and /? not only through 
their trigonometric functions but explicitly. Such goniometric 
considerations, however, were beyond the ken of the Hippo- 
cratean period. 

In the second place, Hippocrates was not concerned with the 
general crescent but with a very special kind, known as 
quadrable. Strictly speaking, a bounded region of the plane is 
quadrable if a square of equal area can be constructed by straightedge 
and compass. Any parallelogram is quadrable, because it can 
be converted into a rectangle of equal area, which, in turn, 
can be converted into a square, the side of the latter being 
the mean proportional between the sides of the rectangle. The 
same holds for any triangle, from which we conclude that 
the area bounded by a general polygon is quadrable, provided that the 
polygon itself can be generated by straightedge-compass operations. 

Another example of a quadrable area is the general parabolic 
segment, i.e., the region bounded by an arc of a parabola and 
the line which joins its end points, for, according to a celebrated' 
theorem of Archimedes, the area of a parabolic segment is equal * 
to two thirds of the area of the triangle bounded by the chord and the 
tangents. Examples of nonquadrable areas are the circle itself, or 
any sector or segment thereof constructible by straightedge 
and compass. 

Thus, the existence of quadrable crescents is anything but 
evident. However, one such crescent is shown in Figure 18, 
and its utmost simplicity suggests that it was known before 
Hippocrates; for all we know, this crescent might have 
served as the point of departure of his investigation. Here the 
outer arc is a semicircumference, the inner a quadrant. A 
direct inspection of the figure shows that the area of the 
crescent is F = a 2 , where a denotes the radius of the outer arc. 



G C 

Some historians maintain that Hippocrates believed that the 
number of quadrable crescents was infinite. I find little evidence 
to support this contention, and even less to justify the sarcastic 
allegation of Aristotle that Hippocrates had invented his 
crescents for the express purpose of squaring the circle. On the 
other hand, it is quite possible that Hippocrates did not rule 
out the feasibility of squaring the circle, and that he used the 
crescents to demonstrate that circular and rectilinear figures 
may possess the same area. This would explain why he limited 
his study to a special variety of crescents which I shall call 
in what follows Hippocratean. 

These special crescents satisfy the following conditions: 
First, the sectors which rest on the two arcs of the crescent, 
AGED and BCFD in Figure 17, are equal in area; and second, 
the central angles of these sectors are commensurable. 



The first of these conditions, with the notations of formula 
(10. i), leads to 

. . . (10.2) 
from which we conclude that 

i.e., that the area of a Hippocratean crescent is equal to the area of 
the quadrilateral bounded by the four radii which pass through its 
"horn-points" Thus, such a crescent is quadrable if, and only if, 
the triangle ABC is constructible by straightedge and compass. 

Applying the law of sines to the triangle ABC, we are led 
to the equation 

sin a 

However, this interpretation does not solve the problem of 
quadrability : on the contrary, it merely accentuates the 
difficulty. The issue is this : is it possible to express sin a and sin /? 
in terms of rational numbers and quadratic surds? This is not a 
question of geometry or algebra, or even analysis; it is a 
problem of transcendental arithmetic, a field of mathematics 
replete with questions that have challenged the keenest minds 
these last two centuries. Some few of these, such as the trans- 
cendental character of the numbers e and TT, have been brought 
to a successful conclusion. However, the solutions were not of 
a type to suggest general methods for attacking kindred prob- 
lems. Indeed, it may be stated without peradventure that 
transcendental arithmetic has more unsolved problems and 
unproved conjectures than any other branch of mathematics. 

Assume next that the angles a and /? are commensurable. This 
means that integers p and q exist such that 

ajp = Pfq or a = poj, ft = qa). . (10.5) 



By substituting these values in (10.4) we put the latter in the 

q sin 2 poj =p sin 2 qa). . . (10.6) 

This last relation may be viewed as the defining equation 
of all Hippocratean crescents, whether quadrable or nonquadrable. 
The integers p and q may, without loss of generality, be 
assumed relatively prime; i.e., angle a* may be viewed as the 
greatest common measure of a and /?. The form of a Hippocratean 
crescent depends only on p and q ; accordingly, I shall denote 
the crescent by the symbol (q, p), where q is the greater of 
the two numbers. 

Equation (10.6) is transcendental in appearance only. We 
shall see, indeed, that it may be transformed by rather simple 
manipulations into an ordinary equation of degree q i. 
If this latter admits of a rational root, or of a root which may 
be expressed in terms of quadratic surds bearing on rational 
numbers, then the crescent is quadrable. If no such "admis- 
sible" root exists, then the crescent is nonquadrable. 

How did Hippocrates cope with these difficulties? How did 
he stumble on the problem in the first place? What prompted 
him to pursue the quest? Was he aware that quadrable 
crescents existed for q 5 but none for q = 4? Did he believe, 
as some commentators insist, in the existence of an infinite 
number of quadrable crescents? Did he restrict the problem to 
what we may call "algebraic" crescents, because he sensed 
that any other crescent is nonquadrable, or was he motivated 
by mere expediency? 

On some of these points we have no knowledge whatsoever ; 
on others the information is most sketchy and often con- 
flicting, which is not surprising, since even the earliest com- 
mentaries on Hippocrates were written many centuries after 
his death. Now, an honest restoration of any work must cling 
tenaciously to the ken and spirit of the period which has given 
rise to the work. In the case of Hippocrates, however, we 



know very little of the ken of the period, and what little we 
do know is largely derived from speculations on his achieve- 
ments. It is a vicious circle, indeed! And yet, so important 
are the Hippocratean methods to the understanding of Greek 
mathematics that I have resolved to risk here a partial 
restoration of his contributions to the problem. 

Hippocrates based his arguments and constructions on certain 
theorems pertaining to similar segments. The term is rarely used 
today, but it played quite an important role in Greek specula- 
tions on squaring the circle. A circular segment is completely 
determined by the radius of the circle and the angle at which 
the chord is viewed from the centre of the circle. If two circular 
segments have equal radii and equal central angles, they are 
said to be congruent. If, on the other hand, they have equal 
central angles but unequal radii, they are said to be similar. 

These statements sound like mere definitions, and yet they 
are more than that. Indeed, the extension of congruence and 
similitude from rectilinear to curvilinear figures which, inci- 
dentally, has been attributed to Hippocrates himself is 
accomplished through an infinite process, the arc of the curve 
being viewed as the limit of a variable polygonal contour. 
By the same token, the metrical aspects of similarity, which grew 
out of the theory of similar triangles and were eventually 
extended to similar polygons, are assumed to retain their 
validity for curvilinear configurations. The circular segment 
is a case in point : the arcs of two similar segments are propor- 
tional to their chords, while the areas of similar segments vary 
as the squares erected on the chords. 

Consider now a crescent of the commensurable type. Denote, 
as before, by 2 a and 2/3 the central angles, by 20) the common 
measure of these angles, and set, as before, a = pa), /? = qo). 
Hippocrates begins by dividing the inner arc into p equal 
parts and joining the points of division by chords; in this 
manner he forms a system of congruent segments, each of 
radius a and central angle 20) (Figure 19). He proceeds in 



the same manner with the outer arc, generating a system of 
q congruent segments each of radius b and central angle 2w. 
It is obvious that the first system of segments lies outside the 
crescent, and it is not difficult to prove that the second system 
lies entirely within the crescent. Furthermore, any segment of 



the first system is similar to any segment of the second, so 
that if we denote by u and v the respective chords of the two 
systems, and by a* and r the areas of the respective segments, 
then, by the lemma mentioned above, 

o- : r = u 2 : v 2 . . . . ( IO -7) 
I 129 


The next step in the Hippocratean reasoning is this: there 
exist, among the infinite variety of commensurable crescents, 
some for which the two systems of segments possess the same 
area, i.e., 

per = qr . . . (10.8) 

Let F be a crescent which enjoys this property; then by deleting 
from the crescent the second system of segments, and adjoining 
to it the first, we form a dosed polygon of the same area as the 
crescent F. The crescent is quadrable, if the polygon is, and the polygon 
is quadrable, if its construction can be accomplished by means of the 
straightedge and compass. (See Figure 19.) 

This reduces the construction of crescent (q, p) to that of a 
polygon of p -f q sides. The polygon has a rather special 
contour. The outer branch is made up of q equal sides each of 
length u and is inscribed in a circle of radius a; the inner 
branch has equal sides of length u and is inscribed in a circle 
of radius b. The sides u and v are connected by the relation 

u :v =Jq : ^p . . . (10.9) 

It is not difficult to show that such "Hippocratean" polygons 
"exist" for any values of the integers p and q. But to state this 
is just begging the question. The issue is: is the polygon associated 
with the crescent (q, p} quadrable? i.e., can it be erected by straight- 
edge and compass? 

As a matter of fact, the area of the Hippocratean polygon 
is equal to the area, A, of the quadrilateral ACBD considered 
in Section 4 of this chapter. For, in virtue of the proportions, 

a : ft p : q and u : v = a : b 

condition (10.9) is equivalent to b 2 oc = a 2 fl, which, as we saw, 
entails F = A. Thus the Hippocratean polygon is constructible by 
straightedge and compass, if the quadrilateral is, and vice versa. 

On the other hand, while the two criteria of quadrability 
are theoretically equivalent, the construction techniques to 
which they lead are far from identical. This may appear as 



of little consequence to one who has all the devices of formal 
algebra at his fingertips and, hence, can pass from one 
approach to the other by a twist of the wrist, as it were. But 
to the classical geometer who had no other way of expressing 
metric relations than in the cumbersome language of a graphical 
algebra, the choice of method was not a mere matter of mathe- 
matical elegance. 



A case in point was the quadratic equation, or rather the 
classical equivalent thereof: a system of simultaneous equations, 
the prevailing types being 

x + y = a, xy = c 2 ; and x y = b, xy c 2 

(10. 10) 

where, of course, a, b and c were viewed as given rectilinear 
segments. The graphical solutions of these two basic problems 
is shown in Figure 20. The mean proportional between two segments 
is the key to both constructions. To us today this means a 
simple arithmetic operation, but Greek mathematicians re- 
garded it as an important geometrical problem, and devised a 
variety of procedures to cope with it. 

One such device, based on a theorem to which Greek 


geometers attached considerable importance, is used in 
Figure 20 for the graphical solution of the simultaneous equations 

x y =J~3> *y = 2 > (io.ii) 

which, as we shall presently see, enter in the construction of 
crescents of type (3.2). The theorem in question may be 
stated as follows : the tangent drawn from a point to a circle is the 
mean proportional between any transversal which passes through the 
point and its external part. Thus in Figure 20: JVT 2 NC.NE. 

In the figure, CE is the side of an equilateral triangle 
inscribed in a circle of radius i, i.e., CE = </~%. The larger 
concentric circle is of radius ^"3, an d the tangent to the smaller 
circle issued from any point, A*, on the larger circle is of length 
^ 2. It follows that the segments NC and NE give the graphical 
solution to (10. n). 


For crescent of type (3.1), the Hippocratean polygon is a 
trapezoid with three sides of equal length. Denoting the latter 
by w, and setting u = i, we find that the fourth side is v = ^3. 
Or, as Hippocrates would have stated this: the greater side of 
the trapezoid is to the lesser side as the side of an equilateral triangle 
is to the radius of the circle into which it is inscribed. 


The construction of (3.1) is, therefore, direct and simple. 
(Figure 21.) On the carrier xx erect MN == i and C'D' = ^ 3 ; 
with M and JV as centres, draw the unit circles (M) and (JV). 
The perpendiculars to xx erected at C f and D' meet these 
circles in the vertices C and D of the trapezoid sought. To 
determine the centre B of the crescent, draw the bisector of 
the angle at M ; to locate the centre A, draw the perpendicular 
CA to CM. 

For crescent of type (3.2), the associated pentagon is a sort of 
truncated trapezoid: CEDNMCin Figure 22. Indeed, by analysing 
the angles of the configuration, we find that the sides of the 
inner branch, CE and DE continued, pass through the vertices 
JV and M of the outer branch. The same analysis shows that 
the sides CM and DJV of the outer branch are tangent to the 
inner arc of the crescent. On the other hand, we have in 
virtue of (10.9), u : v = J~z : ^"3. If then we set u ^~2, 
v = 4/3, we find 

-EN=-CE = V3> CN . JV = JVZ> 2 = 

We are thus led to the simultaneous equations discussed in 
the preceding section and the graphical solution given in 
Figure 20. 


In Figure 22, the order of procedure is: (i) locate the points 
C, E and jV; (2) determine the remaining vertices, M and 
D ; (3) the centre, /?, of the outer arc is the intersection of the 
perpendicular bisectors to the diagonals CJV and DM of the 
trapezoid; (4) the centre, A, of the inner arc of the crescent 
is obtained by erecting perpendiculars to CM and Z)jV. 

The restorations exhibited in Figures 20, 21 and 22 are 
offered here for what they are worth. They are not based on 
any authentic information as to the actual methods used by 
Hippocrates, any more than are the sundry other restorations 
which have been proposed during the twenty-two centuries 
which separate the two periods. All I claim for my own con- 
jectures is that they do not transcend the ken of the period 
during which the problems were solved, as I envisage that ken. 


Returning to the algebraic analysis of the Hippocratean 
problem, I take as defining equation the relation (10.4) 

sin a _ la 
sin/? ~ V /? 

established in Section 4. 

Type (2.1): Here /? 2a. Hence sin 2a = ^2 sin a. 
Eliminating the "trivial" solution sin a o, we arrive at 
cos a = il^~2. Thus a = 45, fi = 90. This is the crescent 
discussed in Section 3 and shown in Figure 18. 

Type (3.1): Here /? = 3 a, and sin 3 a = ^/lj sin a. Replacing 
sin 3<x by 3 sin a 4 sin 3 a, and "shedding," as before, the 

trivial solution, we arrive at sin a = ^3 ^3. The crescent 
is shown in Figure 21 : the angle 7 at C is 2 a. The area of the 
quadrilateral ACBD being ab sin 2 a, the crescent is quadrable. 
We find, indeed : 

T =<z 2 3*2~*. 

Type (3.2): Here a> = 7, a = 27, fl = 37. The equation, 



*J~2 sin 37 = ^3" sin 27, leads to a quadratic in cos 7 = #, 
namely, 4# 2 # ^ 6 i = o. The result is 

cos 7 == (/22 + V6)/8. 

(see Figure 22) 

The crescent is quadrable with 

(4. i) : Here ft = 4<x, and sin 40; = 2 sin a. This leads 
to 4 cos 3 a 2 cos a i = o. If we set sec a = x, we arrive 
at the cubic x 3 -f- 2# 2 4 = o. Any rational root of this 
cubic would have to be an integer and, as a matter of fact, a 
divisor of 4. We find by direct substitution that none of these 
divisors satisfy the equation; thus the cubic has no rational roots, 
and, by the same token, has no quadratic roots. Hence, the 
crescent (4. i) is nonquadrable. 

Type (4.3): Here 7 = o>, a = 30^, /? = 4^. The defining 
equation is ^3 sin 472 sin 37. Proceeding as before, we 
arrive at an irreducible cubic, and conclude that the crescent 
(4.3) is nonquadrable. 

Type (5.1): Here 

sin 5<% = ^/5 sin a 5 sin a - 20 sin 3 a + 16 sin 5 a. 
We are thus led to the biquadratic 

i6# 4 20# 2 + (5 Vs) - 
We find 

= ^ / 


Thus the crescent (5. i) is quadrable. 

Type (5.3): Here a = 36), ft 5^, 7 20). The defining 
equation is: ^3 sin 5^ == Vs sm 3 W * Like type (5.1), the 
latter leads to a quadratic in sin 2 a). The crescent (5 . 3) is, 
therefore, quadrable. 

Types (5.2) and (5.4) lead to irreducible quartzes and are, 
therefore, nonquadrable. 


The preceding discussion follows the lines of an essay published 
in 1840 by the German mathematician, Theodore Claussen, 
who, as far as I know, was the first to subject the Hippocratean 
problem to an algebraic analysis. His analysis, as we just saw, 
yielded not only the three quadrable crescents, (2.1), (3.1), 
(3.2), discovered by Hippocrates, but also the two crescents of 
"order" 5, namely, (5.1) and (5.3); it proved, moreover, that 
the remaining crescents of order 5 were nonquadrable, and 
that the same held for the crescents of orders 4 and 6. Claussen 
conjectured that the Hippocratean problem had no other quadrable 
solutions than the Jive crescents just mentioned; but that if any other 
crescents did exist, they would have to be of prime order. 

This last hypothesis was vindicated when sixty-odd years 
later Edmund Landau proved that not only is the order of a 
quadrable crescent a prime number, but it must be a Fermat 
prime., i.e., of the form 2 n + i. Now, these primes play a very 
important part in cydotomy, i.e., the division of a circle into 
equal parts. In more familiar terms, if it is possible to construct 
by straightedge and compass a regular polygon of an odd 
number of sides, say q, then q is either a Fermat prime, such 
a S3 3 5>i7>257,..,,ora square-free product of such primes. 
Could it be that there was some recondite kinship between 
quadrable crescents and the regular polygons amenable to 
straightedge-compass constructions? 

Well, long before speculations on this theme could gain 
momentum, it was found that none of the 16 crescents of order 17 
was quadrable. The Claussen forecast was further strengthened 
when in 1 934 the Russian mathematician Tchebotarev proved 
that the crescent (q, p) is nonquadrable, if p is odd and q greater 
than 5. Finally, in 1947, the Russian Dorodnov extended 
Tchebotarev 's results to even values of p, thus confirming the 
remarkable conjecture which Claussen had made more than 
a hundred years earlier that the only quadrable Hippocratean 
crescents were 

(2.1), (3.1), (3.2), (5.1) and (5.3). . (10.12) 



Such is the history of a problem which ranks among the earliest 
in the annals of mathematics. Formulated within one hundred 
and fifty years of the Founder's death and which is even 
more significant one hundred and fifty years before Euclid's 
Elements saw the light of day, it remained in a state of suspended 
animation for nearly twenty-four hundred years, and was only 
partly resolved after the combined resources of modern analysis 
and number theory were enlisted in its behalf. 

I say partly, because the Claussen-Landau-Dorodnov theo- 
rems deal with the algebraic aspect of the problem only. There 
is still the question : Are the restrictions imposed by Hippocrates 
necessary conditions of quadrability? Specifically, the problem 
consists of determining values of a and ft for which all three 

sin a, sin ft, and H = ft sin 2 a a sin 2 ft . (10. 13) 

can be simultaneously expressed in terms of rational numbers 
and quadratic surds bearing on rational numbers; or of 
proving that no such values exist, unless H = o. And this problem 
of transcendental arithmetic has, as far as I know, not even been 


Chapter Eleven 


It is one thing to execute a construction by tongue 
as it were, quite another to carry it out with 
instruments in hand. 


The Sophist, Hippias of Elis, a near-contemporary of Hippo- 
crates of Chios, invented the curve for the express purpose of 
squaring the circle. A century or so later, the Hippian solution 
was restored and amplified by Dinostratus, a member of 
Plato's Academy and a brother of Menaechmus of conic 
sections fame. No written records of either the original or the 
restoration exist today. Pappus and other commentators have 
given a good description of the curve itself, but their explana- 
tions as to how the quadratrix was applied by Hippias and 
Dinostratus to the quadrature problem are far from satisfactory. 
Some historians, misled by the term ' 'mechanical" which 
earlier geometers used in describing the quadratrix, insinuate 
that Hippias relied on some sort of "mechanism" to generate 
the curve, forgetting that Greek mathematicians were wont to 
brand as mechanical any construction that implied loci other 
than lines and circles. Other historians, while recognizing the 
geometrical character of the approach, misinterpret the 
motivation, leaving the reader under the impression that 
Hippias and Dinostratus were merely begging the question. 

The problem was to construct a triangle the area of which was 
equal to that of a given circular sector, specifically, a quadrant of a 



circle. In Figure 23 the sector is BOM = %Rs, where R is the 
radius and s the length of the arc. Assume OP equal to s: then 
the triangle BOP is equiareal with the sector. Next, suppose 
that the perpendicular to OP meets the radius OM continued 
in Q: then, as M sweeps the circumference, the point Q 
generates the Hippian quadratrix. If the latter were fully traced, 
it could serve as a templet not only for squaring any circular 



sector, but for rectifying any circular arc. In particular, the triangle 
HOB would be equiareal with the quadrant OAMB, and the 
rhombus HBH'B' equiareal with the circle, and since the 
conversion of a rhombus into an equiareal square is a straight- 
edge-compass operation, the squaring of the circle would be 
effectively accomplished. 

If the Hippian analysis ended here, the Sophist could be 
justly accused of begging the question. For, unless he knew 
how to rectify a circular arc, i.e., to erect a linear segment of 
equal length, he could not generate the quadratrix; and if 
he knew how to rectify a circular arc, then he would not need 
the quadratrix in the first place. However, it is a far more 
reasonable conjecture that, arrived at this point, Hippias 



inverted the problem, i.e., instead of seeking a rectilinear 
segment of length equal to a quadrant of a given circum- 
ference, he sought to determine a circle, one quadrant of which was 
equal to a given rectilinear segment. This meant to construct a 
quadratrix of given base, a task which could be effected 
point by point without stepping out of the traditional domain. 

Q 15 )4 13 12 )! 

10 9 8 7 <5 


4 3 

The procedure is shown in Figure 24. A set of equally 
spaced rays divide the right angle XOT into n equal angles; 
the given segment OH is divided in the same number, n, of 
equal parts. The perpendiculars to OX at the points of division 
meet the corresponding rays in points which lie on the quadra- 
trix sought. By carrying the dichotomy far enough, the set of 
points may be rendered as compact as one wishes. As to the 
vertex, B, of the quadratrix, the goal of the Hippian problem, 
it cannot be reached directly by the operation at hand: it 
should be viewed as the limit of an infinite dichotomy, and this, I 
conjecture, was what Hippias actually had in mind. 



According to this interpretation, the Hippian quadrature was 
an attempt to define the ratio of a circumference of a circle 
to its diameter as a limit of an infinite process. The analytical 
counterpart to this graphical procedure is the formula 

Limit n tan =- . . (n.i) 

2rt 2 v ' 

This, incidentally, was one of the formulae which Archimedes, 
two centuries later, used in determining rational approxima- 
tions to 77. In the Archimedean approach, the circle was 
viewed as the common limit of two series of regular polygons 
of which one was inscribed and the other circumscribed to the 
circle, the number of sides growing indefinitely. When applied 
to a quadrant of a circle, this approach leads to the inequality, 

7T 77 ^ 77 , , 

n sin < - < n tan . . (11.2) 

2H 2 2/2 

To appraise the ideas of Hippias one should remember that 
his quadrature was the first recorded venture into the field of 
infinite processes; that this venture took place about 450 E.G., 
at a time when mathematics was barely more than a century 
and a half old ; that the Hippian quadratrix was the first curve 
(other than the circle, of course) of which we have any historical 
record; that this occurred at least a century before the dis- 
covery of conic sections; and that in his treatment of the curve 
Hippias employed devices which two thousand years later 
were turned by Fermat and Descartes into the basic imple- 
ments of analytic geometry. 

At this juncture I must give a brief outline of an issue which 
is a settled matter today, but had quite an eventful history in 
the past three centuries; namely, the classification of plane 



curves. Greek efforts in this direction, based on strictly geo- 
metrical considerations, were rather superficial and, in a 
sense, sterile. With the advent of analytic geometry, the 
question entered a new phase. 

Descartes was the first to propose that curves be classified 
according to the character of their equations, i.e., of the 
functional relations which represented them in a system of 
rectilinear coordinates. He was particularly interested in loci the 
equations of which could be put in the form of a polynomial 
in two variables. Today, we call such loci algebraic, and define 
as order of the curve the degree, n, of the representative poly- 
nomial. Descartes, however, had different ideas on this sub- 
ject: he defined the order of a locus by the integers \n or 
\(n + i), depending on whether n was even or odd. Thus, what 
we designate today as cubics and quartics were to Descartes 
curves of the second order, while straight lines and conies were 
defined as loci of the first order. 

Newton did identify the order of a locus with the degree of 
its equation, but not without reservations. His reluctance to 
class straight lines among curves led him to define the integer 
n i as the genus of the locus. Thus the straight line was 
catalogued as locus of order i and genus o. In spite of Newton's 
prestige, the idea did not take root, and today the term genus 
is used in an entirely different sense. 

As to such curves as cycloids, spirals and sinusoids, which we 
catalogue today as transcendental, there is no record that Newton 
had a collective name for this non-algebraic variety. He did 
state, however, that the order of such loci should be viewed as 
infinity, inasmuch as a straight line may intersect such a curve 
in an infinite number of points. 

The term transcendental was coined by Leibnitz. This mystic 
philosopher-mathematician was endowed with an extra- 
ordinary intuition and foresight. He anticipated many a 
mathematical trend, sometimes as much as a century before 
it reached the stage of fruition. The division of numbers and 



functions into algebraic, or, as Leibnitz called them, analytic 
and transcendental, is a case in point. As though foreshadowing 
the course of modern algebra, Leibnitz stated, in so many 
words, that to every algebraic number there may be assigned 
one and only one algebraic equation with rational coefficients. 
The degree of this equation he calls the order of the algebraic 
number. Recognizing, however, that magnitudes exist to 
which no polynomial can be assigned, he proposed to call 
these transcendental because, as he put it, they transcend algebraic 

In a similar manner, functions exist which cannot be trans- 
formed into polynomial relations no matter how many rational 
manipulations may be undertaken to that end. These functions 
Leibnitz defined as transcendental. He was frankly puzzled by 
the fact that a transcendental equation may admit of an 
algebraic and even of a rational solution, as, for example, the 
equation x x + x = 30, which is satisfied by the integer x 3. 
However, he viewed such "phenomena" as exceptions, main- 
taining that, "on the whole," transcendental equations are satisfied 
by transcendental numbers only. 

The subsequent course of events has justified Leibnitz' 
perplexity, and has strengthened, at the same time, the plausi- 
bility of his conjecture. The relation between transcendental 
numbers on the one hand and transcendental functions on the 
other persists as an unsolved problem to this day. The histories 
of the Hippocratean crescents, of the transcendence of the 
numbers e and TT, and the many pending questions of trans- 
cendental arithmetic foreshadow that the Leibnitz problem will 
remain on mathematical agenda for some time to come. 

Our textbooks on analytic geometry follow methodically 
the classification just outlined. After disposing of the straight 
line and the circle, they take up the conic sections, or curves 
of the second order. This is followed by cubics and quartics, 
and algebraic curves generally. The study of transcendental 
curves is relegated to the end of the course, and, in some 



curricula, postponed until the student has acquired some 
rudiments of the calculus. 

Now, all this is as it should be. Having put algebra into the 
foreground of mathematical instruction, we must needs begin 
with rational and finite operations; and this predicates the 
order of exposition, whether the subject be number, function 
or graph. And yet, in all fairness to the more sophisticated 
student, we should inform him that historical sequence has 
not always agreed with the order of exposition of a textbook. 

The Hippian quadratrix is a case in point. Choose the base 
of the curve and its axis of symmetry for reference frame, and 
the radius of the generating circle for unit of length. Then, by 
definition, arc BM = OP = x ; again, if we agree to measure 
angles in radians, then agle BOQ is also equal to x, and the 
right triangle POQ, yields the relation: 

y = X COtX . . . (ll.3) 

This is the equation of the curve in rectangular coordinates, 
and inasmuch as cot x cannot be represented as a polynomial in x, 
the quadratrix is a transcendental curve. 

Thus, after the straight line and the circle, the first locus 
studied by mathematicians was a transcendental curve, while even 
such simple algebraic curves as the conic sections did not 
emerge on the mathematical scene until a century and a half 
later. Truly, history is no respecter of systems. 


Chapter Twelve 


Magnitudes are said to have a ratio to one another 
when the lesser can be multiplied so as to exceed 
the greater. 


Book Seven of the Elements contains the description of a 
numerical device which has come to be known as Euclidean 
algorithm, although it probably antedates Euclid by at least 
one hundred years. Euclid applies the algorithm to deter- 
mining the greatest divisor common to two integers. To illustrate, 
take the numbers 2,601 and 1,088; a chain of successive 
divisions leads to the identities: 

2,601 = 2 X 1,088 + 425 

i, 088 = 2 X 425 + 238 

425 = i x 238 + 187 

238 = i x 187 + 5 1 

187 =3 X 51 + 34 

51 = i x 34+17 

34 =2 X 17 

The last residue, 17, is the greatest common divisor sought, 
because any divisor common to 2,601 and 1,088 must also 
divide the remainder 425, and, by the same token, must 
divide 238, 187, 51, 34 and 17. 

When the same algorithm is applied to two relatively prime 
integers, A and B, it results in the expansion of A/B into a 
regular continued fraction. I shall illustrate this in the case of 

41 = 16 x 2 + 9 or 41/16 = 2 + 9/16 

16 = 9x1+7 or 1 6/9 =1+7/9 

9 == 7 x i + 2 or 9/7=1+2/7 

7 = 2 x 3 + i or 7/2=3 + 1/2 

K 145 


Hence the expansion: 

41 , i 

-_ = 24- 

16 i + 

1 +1 

3 + 1 


Observe that the quotients generated by the algorithm become 
the terms of the expansion. Since the continued fraction is 
completely determined by these quotients, we may write 
without ambiguity: 41/16 = (2; i, i, 3, 2). 

It is obvious enough that the procedure exhibited on this 
example applies to any two integers. Thus any rational number 
may be expanded into a terminating continued fraction, and 
the expansion is unique. By the same token, any rational number 
can be represented by a finite array of positive integers. 
In what follows I shall call that array the spectrum of the 

The expansion into continued fraction, the Euclidean algo- 
rithm and the spectrum to which it gives rise are susceptible 
of a striking graphical representation which is exhibited in 
Figure 25 in the case of 41/16, the example treated above. 
In the graph, the numerator and denominator of the fraction 
are interpreted as sides of a rectangle. From this rectangle 
we remove as many squares as possible, leaving a residual 
rectangle to which the algorithm is applied anew, and the 
process is continued until no residual rectangle remains. The 
number of squares in each "tier" gives the corresponding 
element of the spectrum, i.e., a term of the continued fraction. 
I said that to any rational number corresponds a unique spectrum. 
Conversely, any ordered array of positive integers may be construed 
as the spectrum of some rational number. To calculate the latter, 
one could express the spectrum as a continued fraction and 
then reverse the Euclidean algorithm. A far more effective 
method was discovered by John Wallis. The Wallis algorithm 
is exhibited in the following table for the spectrum (2 ; i, i, 3, 2). 



Terms of Spectrum 






Numerators of 



3X 1 + 2 = 5 

5X3+3 = i8 

18X2+5 = 41 

Denominators of 



IX I-f 1=2 

2x3+1= 7 

7X2 + 2=l6 

The successive conver 'gents are the values of the "curtailed" 
spectra, the last convergent being the value of the continued 
fraction. Thus, the Convergents of 41/16 are 

2 3 5 l8 and t 1 
_, _, _, _ and i6 . 






The theoretical importance of the Wallis algorithm is that 
it opens the way to a rigorous treatment of infinite continued 
fractions. To be sure, such infinite processes were used long 
before Wallis: explicitly, by the Italian mathematicians of 
the sixteenth century; implicitly, by Fibonacci, Hero and 
Archimedes. However, Wallis, and later Huygens, put the 
theory on a solid basis by establishing that the process was con- 
vergent for any infinite spectrum, i.e., that the sequence of conver gents 
always approached a limit. This limit is necessarily irrational, since 
the spectrum of any rational number is necessarily finite. 

We conclude that any array of positive integers, finite or 
infinite, may be interpreted as the spectrum of some real 
number F : if the spectrum is finite, F is rational; if the spectrum 
is infinite, F is irrational. Conversely, any positive number F, 
rational or irrational, may be expanded into a continued frac- 
tion : the expansion is finite, if F is rational; infinite, if F is irrational. 
Furthermore, the individual steps in the expansion of an 
irrational number into a continued fraction follow the pattern 
of the Euclidean algorithm, and the infinite algorithm may, accord- 
ingly, be viewed as a direct generalization of the finite. 

The generic operation of this extended algorithm can be expressed 
in very simple terms, by introducing the symbol [F] to denote 
the greatest integer contained in the positive number F. Discard the 
trivial case when F is an integer; then, the difference F [F] 
is contained between o and i, which means that there always 
exists a number X greater than i such that 

F=[F] + ~. . . . (12. i) 

If F is a rational number, say A\B, then [F] is the quotient 
in the division of A by B, and the operation is but a paraphrase 
of a step in Euclid's algorithm. If, however, F is irrational, 



then X, too, is irrational. Operating on X as we did on F, 
we obtain X = [X] -\- i/T, where T, too, is an irrational 
number greater than i. The process will, therefore, continue 
indefinitely, generating an infinite sequence of positive integers 

L m, [.a, ..... (12.2) 

These are the denominators of the infinite continued fraction which 
tends to the irrational F as limit. 

The Dutch mathematician Huygens is credited with being the 
first to use continued fractions as a means of deriving rational 
approximations. The effectiveness of the method rests on two 
properties of the Euclidean algorithm which may be formulated 
as follows : 

I. If A/B and A'/B' are two consecutive convergents in the 
expansion of the number F, then F is contained between A/B 
and A'/B'. Thus, we shall see in the sequel, that 265/153 and 
362/209 are the ninth and tenth convergents respectively in 
the expansion of -^3, which means that 

153 ^ 209 

II. If A/B is a convergent in the expansion of F, then the 
error committed in writing A/B for F is less than i/B 2 . Thus, 
by taking 362/209 for ^3; we are approximating the latter 
within an error less than 1/40,000. As a matter of fact, com- 
paring the fraction with the tabular value of ^3, we find: 

A/3 = i-TS^S 1 - - > 362/209 = 1-732052 . . . 

Unfortunately, the effectiveness of the method is largely 
vitiated by the tedious work involved in determining the 
spectrum of the irrational. In this respect, the irrationals of 
the type M + ^ JV", where M and JV are rational numbers, 
are in a class by themselves, in that the pattern of the spectrum 
can to some extent be predicted beforehand, as may be seen 
from the following examples. 



First example; the golden section ratio. The reciprocal of the 
"divine proportion" is F = (V~5 + i). (See Chapter Five.) 
We find [F] = i and T [F] = |(V5 0- The reciprocal 
of this is X = J(/5 + i) ; thus Jif == F and [-Y] = i. Hence 

KVs + *) = (*; 1, i, i, - - - ) - ( I2 -3) 

The Wallis algorithm yields here a so-called Fibonacci sequence : 
i, 2, 3, 5, 8, 13, 21, 34, . . . , . (12.4) 

where any term beginning with the third is the sum of the two terms 
which precede it. 

Second example. Let us seek the expansion of the quadratic 
surd F = ^3, which will play an important part in the dis- 
cussion of the Archimedean approximation. We find here 
r [F] = V3 i, the reciprocal of which is X == J(V3 + 0- 
Hence, X - (X) = (Vs - i), T = Vs + i. Thus, 

The reciprocal of the latter is 4(\/3 + !)> i-c., ^, and we 
conclude that 

V3~ = (i : 1,2, 1,2, 1,2, . . . ). . (12,5) 

TAzrrf example. Set F = ^/I9- The procedure is the same as 
in the preceding examples. I leave the details to the reader. 
The result is : 

i9 = (4; 2,1,3,1,2,8, 2,1,3,1,2,8, 2,1,3,1,2,8, . . . ) . (12.6) 

The expansions in the preceding examples have one feature 
in common : each contains an infinite number of identical blocks 
of terms. Such continued fractions are known as periodic; the 
recurrent block is called the cycle, and the number of terms in 
a cycle the period of the expansion. Thus, for example, in the 



expansion of ^Ing the cycle is 2,1,3,1,258 and the period 
is 6. 

The very procedure used in deriving these expansions 
suggests that, when applied to irrationals of type M + *fN, 
the Euclidean algorithm will invariably generate a periodic 
spectrum, and such is indeed the case: the spectrum of any irrational 
of the form M + V~N* where M and N are rational numbers, is 
necessarily periodic; conversely, the limit of any periodic continued 
fraction is a root of some quadratic equation with rational coefficients, 
i.e., an irrational of the binomial type M + V"N- 

There is a remarkable analogy between periodic continued 
fractions and periodic decimal fractions. If T is a positive rational 
number, i.e., the root of a linear equation with rational coeffi- 
cients, then the decimal fraction which represents F is either 
terminating or periodic. Similarly if F is a positive root of a 
quadratic equation with rational coefficients, then the continued 
fraction which represents F is either terminating or periodic. 
In the linear case the generating process is long division, in 
the quadratic it is Euclid's algorithm. 

These periodic properties of quadratic irrationals were 
known to Huygens, Wallis and even to Bombelli, the Italian 
mathematician of the sixteenth century who was the first to 
use continued fractions explicitly. Euler and Lagrange not 
only provided these theorems with rigorous proofs, but showed 
how periodic continued fractions may be used to attack 
some difficult number-theoretical questions. The field was 
vastly extended by Legendre, Gauss, Jacobi, Galois and 
Liouville, among many others, until today a comprehensive 
exposition of the theory would require many a volume. With 
all that, quite a few questions propounded by these masters 
remain unanswered. Chief among these is the relation between 
the character of an integer N and the period and cycle of the spectrum 

Chapter Thirteen 


Indeed, even more important than safeguarding 
truth is the preservation of the methods which have 
led to its discovery. 


In a tract entitled Cydometry, Archimedes made use of the 

Both fractions are excellent approximations to ^3, and it was 
this precision that had enabled the master-calculator to 
evaluate the ratio, n, of the circumference of a circle to its 
diameter within the narrow limits : 

310 ^ 10 , ^ 

<7T < 4 . . . (13.2) 

71 ^70 

Archimedes described in detail the successive steps in his 
evaluation, but gave no inkling as to how he had arrived at 
the approximations to ^3 which had served as his point of 
departure. Could it be that these values were such common 
knowledge among the geometers to whom the tract was 
addressed that he viewed such elaborations as redundant? 
Perhaps! Still, speculations have been rife ever since as to 
the motives which had governed the Archimedean choice. 

Both approximations are convergents into a continued frac- 
tion for ^3, and so it is natural to suspect that it was the 
Euclidean algorithm that had led to these values. However, 
this conjecture has been contested by historians on the ground 
that continued fractions were not introduced until the sixteenth 
century, that the theory did not come to full fruition until the 



eighteenth, and that it was, therefore, entirely outside the 
ken of Greek mathematics. This last statement deserves closer 

We know that the Euclidean algorithm was born on Greek 
soil, that there is nothing in its definition or execution which 
would restrict it to rational numbers, and that men like 
Eudoxus, Euclid and Archimedes could not have failed to 
recognize in it an ideal criterion of commensurability of two 
magnitudes. I would not go so far as to assert that the algo- 
rithm was invented for the express purpose of defining irra- 
tionals, but it is not unlikely that whoever discovered the 
process was at the time in quest of such criteria of com- 
mensurability. To be sure, no documentary evidence exists to 
substantiate this conjecture; but then one should remember 
that Greek geometers studiously avoided the use of such terms 
as infinite or limit. 

A case in point is the Archimedean theorem on the area of 
a parabolic segment, the proof of which depends on the summa- 
tion of the infinite geometric progression 

At no time did Archimedes state that the sum of the progres- 
sion approached 4/3 as a limit, or words to that effect; he 
merely maintained that no matter how many terms were 
taken, their sum would never exceed 4/3. 

It has also been contended by some historians that even if 
some Greek mathematician had envisaged the possibility of 
applying the Euclidean algorithm to quadratic surds, he 
would have lacked the requisite technique to implement it. 
This argument, too, is unfounded. Indeed, Book Ten of 
Euclid's Elements presents a comprehensive theory of binomials 
of type A + \A> including all the operations necessary for the 
expansion of such irrationals into continued fractions. 



However, there is one aspect of the Archimedean approxima- 
tion that casts reasonable doubt on the continued fraction 
conjecture. We saw in the preceding chapter that the expansion 
of ^3 leads to the periodic spectrum (i; 1,2, 1,2, . . . ). The 
The first twelve convergents of this expansion are : 

^ l 
4 l 


= 2 


F K = 




= - 5 

= 362 


= L.35 1 

Observe that while the two fractions which enter in inequality 
(13.1) are among these convergents, they are not consecutive 
convergents. To determine F 12 by the Euclidean algorithm one 
would have to calculate F n first. But if Archimedes had F i: 
at hand, why did he not use it to obtain the "sharper" inequal- 


i* < M5E? 
V3 < 780" 

A passage from Hero's book Metrica offers a clue to this riddle. 
This work contains, among other valuable historical material, 
the celebrated area formula., which is the theme of the next 
chapter. The application of this formula requires, generally, the 
approximate evaluation of a square rooty and so Hero takes time 
out to instruct his readers how to perform such an operation 
"with accuracy and dispatch." The Heronian algorithm is a 
precedure which one would class today as successive linear 
interpolations. While Hero used it for the extraction of square and 


cube roots, the method is far more general, and, as a matter of 
fact, can be successfully applied to the solution of the equation 
F (x) = 0, where F (x) is any single-valued rational function. 

In Figure 26, the curve (F) is the graph of the function 
y = F (x). The abscissa a of the "pole" A is some rational 
number "reasonably" close to the true value of the root x] b is 


another such rational number, and B the corresponding point 
on the curve (F) . The line A B meets the #-axis in a point of 
abscissa c : the crux of the principle is that c is a better rational 
approximation to x than either a or b. By performing the same 
operation on C, we generate a rational number d which 
approximates the irrational x even better than c. It follows 
that by repeated application of the algorithm we may approach 
the root x with any desired precision, the rapidity of the conver- 
gence depending largely on the choice of the pole A. 



For the evaluation of JR, the algorithm leads to the iteration 

As an illustration, consider the example treated in the Metrica; 
^720 = 1 2^/5". Set a 2 : then 

+ 2 

With the initial value u : = a = 2, we arrive at the set 

q 38 161 

1 = 2,,=-,3=-,4= , 

and the last term yields for ^"720 the approximation 161/6, 
which is the result given by Hero. 

The extraction of 3 R leads to the iteration formula 

As an example, let us evaluate 3 *fio. Here J? = 10, and we take 
for a the greatest integer contained in %/io Thus 


Set Wj = a = 2 : then M 2 = 13/6 and u 3 1010/469, which 
deviates from the tabular value of V 1 ^ 2.1536, by less than 

Let us now return to the Heronian technique for the extrac- 
tion of square roots and examine how it works out in the case 
of ^3. To this end, set in formula (13.5) R 3 and denote 
by u n and w rt+1 any two consecutive terms in the iteration 
process ; the result is the relation : 



**>-? - <- 8 > 

As I said before, the sequence will converge to ^3 for any 
reasonable choice of the initial value, G. It is obvious, however, 
that the choice of the operator G will have a decided influence 
on the rapidity of the convergence. For reasons, on which I 
shall not speculate here, Hero chose a = 5/3. This leads to 
the formula, 

and to a sequence, the first four terms of which are the 

5 26 2615 1,3*)! / \ 

a 2 _po . . (13.10) 

3' 15' 153' 780 ^ J 

Now, the last two are precisely those which enter in the 
Archimedean inequality for ^3. Was this coincidence accidental? 
Hardly! To be sure, in describing his procedure, Hero makes 
no reference to Archimedes or to any earlier sources for that 
matter. And yet, it is utterly inconceivable that a brilliant 
scholar who had full access to the Alexandrian Library would 
be unfamiliar with such an important work as Cyclometry. It 
is far more probable that the technique described in Metrica 
was not a Heronian discovery, but was known to Archimedes 
and even to his predecessors. 

Indeed, the ideas back of the procedure are fully within the 
ken of classical mathematics. Thus, the individual steps in the 
process are based on linear interpolation, which, after all, is the 
most natural approach to the technique of approximation. 
As to the process itself, one makes a "reasonably good guess" 
and iterates the guess on the theory that the result is closer to 
the true solution than the original value. The scheme was 
widely practised in classical days, and was described in Latin 
textbooks on mathematics as regulafalsi. 



The four Heronian approximates to ^3 listed in (13.10) are 
all convergents in the expansion of ^3 into a continued fraction. 
What is more, it can be shown that the infinite sequence generated 
by the iteration (13.9) is made up entirely of such convergents. In 
fact, if we denote, as before, by F n the convergent of rank n, 
the terms of the infinite sequence can be expressed as 

When viewed in this light, the Heronian algorithm appears 
as a sort of a "hop-skip" scheme, the net effect of which is 
the speeding up of the Euclidean algorithm. And this is not 
all : the iteration used by Hero is but a special application of a 
general property contained in the relation 

F } = ,l = p 

9 r n) _ 7- * 


To illustrate, consider F 12 . In virtue of (13.12) we can 
evaluate this convergent in several ways. Thus 

(A) F 12 = 


19 X 41 + II X 71 1,560 


/T>\ i? uf c 1 i? \ ^ ^ ^ i 3 ^ ^5 ^ ^5 ^?35^ 

(r>) r I* =Jn.[r^rQ) = - ^ = ^~ 

2 X 15 X 2O 780 

Furthermore, we can derive the 24th convergent from the 
1 2th without intermediate steps, since 

2.F 12 2.1,351.780 

The denominator of this convergent is of the order of io 6 ; it 
follows that the fraction approximates ^"3 within an error 
less than io- 12 . 



The remarkable kinship between the Euclidean and Heronian 
algorithms is a consequence of a theorem on periodic continued 
fractions due to Lagrange. The proof of this fundamental 
proposition is beyond the scope of this volume. I shall add, 
however, for the sake of completeness, that the Lagrangian 
theory covers the most general surd of type *JR 9 where R is a 
positive integer. As a result, the relationship between the two 
algorithms can also be extended to the most general surd of 
that type. Indeed, even the outward form of this kinship is 
retained. For, within some reservations on which I shall not 
insist here, the formula 

H(F m) F n ) = ?^* R - - F m+n . (13.13) 

^m + *n 

remains valid for the corner gents of the general surd *JTl. 

The Lagrangian memoir on the subject appeared about 
1775. Thus more than two thousand years separate the two 
episodes of the story I have told here. The arguments and 
technique used by Lagrange in establishing his theory, and the 
consequent kinship between the two procedures, involve con- 
siderations of algebra and number theory which were alto- 
gether outside the ken of Greek mathematicians. Granted that 
the alleged calculating technique which had led to the inequality 
did not transcend their mathematical knowledge, there still 
remains the perplexing question as to what governed their 
choice of the initial value, 5/3. 


Chapter Fourteen 


The mathematician is like a Frenchman : you tell him 
something, he translates it into his own language, 
and at once it becomes something altogether different. 


We know as little about the life of Hero as we do of Euclid's 
life; in fact, less. For, while we can definitely place the 
Elements about 300 B.C., all we can say about Hero is that he 
flourished in Alexandria during the first century. Unfortunately, 
there were two first centuries, and the question whether his 
activities belong to B.C. or A.D. has not been settled to date. 
Of one thing we are certain : Hero's Metrica and Dioptra ante- 
date the Almagest of Ptolemy, who flourished about A.D. 150. 

The celebrated Heronian formula is usually presented in 
the form 

T = V^^X^Afr""') ('4- 

where a, b, c are the sides, T is the area of the triangle and 
s = %(a + b + c) the semiperimeter. To prove his theorem Hero 
takes the triangle a = 7, b = 8, c = 9. Here, s = 12, 
55=5, s b = 4, s c = 3. Hence T = V?2O, which 
Hero evaluates by the method described in the preceding 

The most natural approach to establishing the formula is 
to start with the theorem that the area of a triangle is one 
half of the product of base and altitude, and this is how most text- 
books proceed. To calculate the altitude, the Pythagorean 
theorem is applied, which brings in the segments of the base, 
and these are later eliminated by rather complicated algebraic 
manipulations. Such a proof is usually accompanied by the 
apology that it is not difficult, just involved. But elementary 

1 60 


as this approach may appear to us today, it was entirely 
beyond the ken of the Heronian period. 

In the Heronian proof, the auxiliary element is not an altitude 
of the triangle but the radius of the inscribed circle. In Figure 27 
the centre J of the inscribed circle is joined to the vertices 
A, B, C. The three triangles BJC, CJA, AJB have areas 


|ra, \rb, \rc> respectively, where r is the radius of the inscribed 
circle. Hence 

T = %r(a + b + c) = rs. . . (14.2) 

Thus, the problem is reduced to expressing r in terms of a, b, . 
To this end, Hero introduces an escribed circle, which touches 
the sides CA and CB on the outside, and the side AB between 
A and B. He then proceeds to evaluate and this is the crux 
of his proof the tangential segments, in terms of the sides. It is 
easily shown that 

= CV = s-c, AV =s-b, CV = s. 

Once this is established, the rest is a matter of similar triangles. 
The centres J and E being on the bisector of the angle C, the 
triangles JVC and EV'C are similar : hence, if we denote the 
radius of the escribed circle by /?, we have 

p : r = (s c} : s. 

On the other hand, JAE is a right angle, and, consequently, the 
right triangles JVA and EVA are also similar, i.e., 

r : (s a] = (s b) : p. 
Eliminating p between these two relations, we arrive at 

and combining this with (14.2), we obtain the Heronian 

Hero illustrates the method on a score of examples. In each 
case the sides of the triangle are integers. He is fully aware 
that the area of such a rational triangle is generally an irrational 
number. However, he states without proof the existence of an 
infinitude of triangles with rational areas as well as sides. 
Today we call such triangles Heronian triples. Any Pythagorean 
triple is obviously Heronian, since T = \ab. We shall presently 



see that by a proper combination of two Pythagorean triples 
one can generate at least one Heronian triple. 

The construction is shown in Figure 28, where ABC and 
A'B'C' are two Pythagorean triangles, and %X and %Y are 
parallel to AB and A'B', respectively. Let us take for h = 


8*8 64 


-64 + 35-99- 

FlGURE 28 


the least common multiple of the integers b and b' : we can then 
write h Jib N'b', where JV and JV' are relatively prime. 
From similar triangles we deduce the sides of XY as 

y = 

= Na + N'a'. 


The triangle is Heronian, because its area T = \hz is a rational 
number. In Figure 31 the Pythagorean triples are (8, 15; 17) 
and (7, 24; 25) and the reader will verify that the resulting 
Heronian triple is (125, 136, 99). 



A far more elegant solution to the Heronian problem was 
given by Brahmagupta, a Hindu mathematician of the seventh 
century A.D. He takes for parameters the ratios, a, /?, 7, defined 


a = = cot $A, /? = = cot \B, 

j = i^f = cot \C. 


Assuming that A and B are the acute angles of the triangle, 
the parameters a and fl are greater than i ; as to 7, it is greater, 
equal or less than i, depending on whether C is acute, right 
or obtuse. On the other hand, a, /?, 7 are not independent of 
each other. They are connected by the relation: 

afly = a + ft + y or y = -^-, . (14.6) 
which reduces the problem to two degrees of freedom. 

These considerations are valid for any triangle; but in the 
case of a Heronian triple, the radius r is rational, and, conse- 
quently, the parameters a, /?, 7 are rational numbers. Conversely, 
if the sum of three rational numbers is equal to their product, and two 
of these numbers are greater than i , then they can be taken 
as Brahmagupta parameters of a Heronian triple. As a matter of 
fact, in so far as form alone is concerned, the triple is completely 
defined by the proportion 

To illustrate, take a = 2, /? = 7/4. We find 7 = 3/2. 
Bringing these to a common denominator and dropping the 



latter, we are led to the integers 8, 7, 6. Hence the triple 
(7 + 6> 6 + 8, 8 + 7), i.e., (13, 14, 15). As another example 
take a = 3, /? = 7; then y = , which means that the angle 
C is obtuse; proceeding as before, we find the integers 6, 14, i 
which lead to the triple (15, 7, 20). 

The elegance of the Brahmagupta approach is further 
accentuated by the fact that the aggregate of Pythagorean 
triples is included under y = i . Indeed, if we set a = p/q, then 
ft has the value (p + q)l(p <?); and equations (14.7) lead to 

It follows that p and q are the Fibonacci parameters of the 
Pythagorean triple. (See formula (9 . 7) of Chapter Nine.) 

Among the Heronian triples treated in Metrica was the set 
of consecutive integers (13, 14, 15). Hero found other such 
"consecutive" triples, and, probably, surmised that their 
number was infinite. However, he failed to devise a systematic 
procedure for generating such triples. Nor did the Hindu and 
Arabic mathematicians fare better in this respect. In modern 
times, the quest resulted in an interesting and rather important 
development in higher arithmetic. Like so many other chapters 
in the history of number theory, this one, too, began with 

Observe that the middle term of a consecutive Heronian 
set must be even, for, otherwise, the semi-perimeter, s, of the 
triple would not be an integer. Accordingly, we set b zx, 
and deduce: 

a = <2x i, b = 2#, c 2x + !) 2s = 6x 

s a = x -}- i, s b = x, s c = x i, ^= 



The problem is, therefore, to choose x in such a way that 
3 (# 2 i) be a perfect square; or which amounts to the same 
thing such that x* i be of the form 3jy, where y, too, is an 
integer. We are thus led to determine all integral solutions of 

x* 3? 2 = i . . . (14. 10) 

Equations of this type are known as Pell equations. Their 
general form is 

x 2 Ry 2 = i . . . (14.11) 

where R is a non-square integer. The strange thing is that John 
Pell, the obscure British mathematician after whom the equa- 
tion was named, had nothing to do either with the formulation 
or with the solution of the problem. It was proposed by Fermat 
shortly before his death as a challenge to British mathemati- 
cians. We have no record of Fermat's own solution, and the 
one attributed to Wallis is far from satisfactory. As opposed to 
this, the solution given by Lagrange a hundred-odd years 
after the problem was proposed is a model of elegance, simpli- 
city and rigour. 

Lagrange derives the solution of the Pell-Fermat equation 
from a few elementary properties of binomials of type 
x +J>^R, where the ' 'modulus, 5 ' /?, is a non-square integer, 
while x and y are any two rational numbers. In particular, 
x and y may be whole numbers, and this is what we shall 
assume in what follows. It is convenient to present these 
properties in the form of lemmas : 

LEMMA A. The product of two binomials of modulus R is a binomial 
of the same modulus. In symbols 

(A) ( X +y4R}(x' +/V*) = (**' + Ryy'} + (*/ + * V* 

LEMMA B. By repeated application of lemma A we find that 
the power of a binomial of modulus R is a binomial of the same 
modulus. In symbols, if n is any positive integer, then 

(B) (x+^Ti)=X+rjR 

where X and T are whole numbers, provided x and_j> are. 

1 66 


LEMMA C. If two conjugate binomials be raised to the same 
power, n, then the resulting binomials are also conjugate. In 
symbols : 

(C) (x + ^~R} n = X + TjR entails (x -yjR) n = X - TjR 

LEMMA D. By combining these properties, we conclude 

(D) if (x + JS/7Z)" = X + YjR, 
then (x 2 - Ry*) = X 2 - R T* 

This last lemma permits one to derive an infinite number 
of solutions to a Pell equation, if one solution is known. 
Assume, indeed, that p and q are two integers which satisfy 
the Pell equation of modulus R, i.e., that p 2 Rq 2 = i ; next 
consider the infinite sequence of binomials 

P 2 + q*jR =(P + 9jR)*>P* + W* =(P + W*) 8 > etc - etc - 

. (14-12) 

Then, in virtue of Lemma D, 

p a *-Rq t * = I, p* -Rq* z = I, p n * -RVn* = I, . . . 

Thus, the sets p 2 , q%\ p& q^ . . p n , q n \ are also solutions 
of the Pell equation of modulus R. 

To illustrate, the equation x 2 7j 2 = i is satisfied by 
* z=8, _>> =3. We find (8 + 3V7) 2 = 127 + 48^/7. Thus 
x = i27,jy = 48 is also a solution, as may be verified directly. 

Thus, the equation x 2 Ry 2 = i admits of an infinity of 
solutions if it admits of one. But with this the problem is by no 
means settled. Two questions remain: First 9 do solutions 
always exist? Second, if they do exist, can the Lagrange algorithm) 
(14.12) generate all solutions from a single basis? Through a 
subtle reasoning, which has later been simplified somewhat 
by Gauss and Dirichlet, Lagrange proved that a basic solution 
always exists, and that it is unique, for any value of R. 

To determine this basic, or minimal, solution for any given 
R is quite another matter. Thus in the case of R = 13, the 
minimal solution is x = 649,^ = 180, while for R = 94, x and 
y contain 6 digits each. However, the quest can be greatly 


facilitated by the following fundamental theorem due to 
Euler : 

If X and y are solutions of the equation, x 2 Ry 2 = i, then x/y 
is a convergent in the expansion of ^ R into a continued fraction. 

Let us now return to the problem of determining all con- 
secutive Heronian sets. We saw in section 6 that the question leads 
to the Pell equation 


= i 

where 2x is the middle term of the triple. The minimal solution 
of this equation is, obviously, # = 2 j> = i. The Lagrange 
algorithm yields the binomials 

( 2 + ^3)2 = 7 + 4 ^ ? ( 2 

The table below gives the results for the first six triples. 


r = y 


b = 2* 


r= 3 *j, 













* r 5 



























It should be noted that the parameter y gives the radius of 
the inscribed circle. The last column shows the connection 
between these Heronian triples and the comer gents to ^3, listed 
in the preceding chapter. 

1 68 

Chapter Fifteen 


Truth : a brief holiday between two long and dreary 
seasons, during the first of which it was condemned 
as sophistry and during the second ignored as 


To one unfamiliar with the idiosyncrasies of Greek mathe- 
maticians the story which I am about to unfold will sound 
quite unreal. Yet, it is just one more confirmation of the 
proposition that history is no respecter of order. 

In an earlier chapter I examined the conjecture that Baby- 
lonian learning had in some way penetrated into Greece 
during the formative stages of geometry, a hypothesis recently 
advanced to explain its prodigious progress in the three pre- 
Euclidean centuries. I rejected the conjecture on the ground 
that the clay tablets thus far deciphered give no more evidence 
of deductive reasoning than the Egyptian papyri did. 

As a matter of fact, Babylonian learning eventually did 
infiltrate Greek thinking, but by then classical geometry had 
been all but consummated, even as the glory of Babylon. 
Indeed, the Greek who sponsored this momentous event was 
the astronomer Hipparchus who flourished about 150 B.C., 
that is, at a time when Apollonius was already a mere memory, 
Archimedes a legend and Euclid ancient history. As to Babylon, 
the one lasting monument of its glorious past was the collection 
of star calendars compiled by Chaldean priests and extending 
back to times immemorial. 

Hipparchus took full advantage of these Babylonian records 
when he wrote his treatise on astronomy. The book was 
subsequently lost, as was a similar manual by his follower 
Menelaus. It is believed, however, that the essential features 



of both treatises were incorporated in Ptolemy's Almagest 
which appeared about A.D. 150. Unfortunately, the Greek 
original of the Almagest has also been lost, and such parts of 
the work as have reached us are in the form of Latin transla- 
tions of Arabic versions; and there is no telling how much of 
their own the zealous Arabic commentators had added to the 
Greek text. That these suspicions are fully justified is suggested 
by the very word Almagest, which is compounded of the Arabic 
article al and an exotic syncopation of the Greek title of the 
book Megale Syntaxis, i.e., Grand Compendium. 

Under the circumstances, it is more than likely that many 
of the innovations, whether mathematical or astronomical, 
which have been attributed to Ptolemy were the achievements 
of other men. Thus, there is no doubt that it was Hipparchus 
who first adapted the Babylonian system to the measurement 
of both time and angles. These sexagesimal units were later 
latinized as gradi, partes minutae primae and paries minutae 
secundae; these, in turn, became the degrees, minutes and seconds 
which, for better or for worse, have survived to this day. 

We are confronted here with the perplexing fact that Greek 
geometry was a finished product when the first chapter of 
trigonometry had not even been written. Nor, for that matter, 
was this first chapter ever written in Greek. The Almagest 
contained a comprehensive treatise on spherical trigonometry, 
but neither Ptolemy nor his Hindu and Arabic successors ever 
referred to any textbooks on plane trigonometry. 

Now, Hipparchus, Menelaus and Ptolemy were astro- 
nomers, and their interest in spherics is quite understandable. 
What is bewildering is that these men had, apparently, plunged 
headlong into spherical trigonometry without using plane 
trigonometry as an intermediary. And yet, after all, whenever 
three points on a sphere define a spherical triangle, they also 
define a plane triangle, and it is the study of the relationship 
between these two species the Greek tripleuron versus trigonon 
that leads to the laws of spherical trigonometry. 



Upon closer scrutiny we find that Ptolemy did make use of 
what we call today plane trigonometry, but failed to honour it 
with a title. Indeed, he constantly invoked the theorems and 
constructions of the classical geometry of triangles, as Euclid 
had taught it; and, in the last analysis, the formal laws so 
prominently displayed in our textbooks are but artful para- 
phrases of these classical propositions in terms of trigonometric 

Thus, as I pointed out in the chapter dealing with the hypo- 
tenuse theorem, the so-called law of cosines was but a form 
of the extended Pythagorean proposition. The law of half- 




angles is a sort of numerical counterpart of a property of the 
inscribed circle, and was, as we saw, implicitly used by Hero 
in deriving the formula which bears his name. As to the law of 
sines, it calls for a more detailed discussion, because the sine was 
a basic concept in the Ptolemaic approach to trigonometry. 

Indeed, the sine was the only one of the six trigonometrical 
ratios which the Greeks had honoured with a name. It was 
called XopS 7 !) i- G -> tne chord. Back of this designation was the 
inscribed angle theorem which I have already mentioned on 
several previous occasions. In Figure 290, UW is a fixed chord 
of a circle of diameter D ; the vertex V of the inscribed angle 
is free to move along the arc subtended by the chord, but the 



magnitude of the inscribed angle remains constant throughout 
this motion. 

It follows that this magnitude is fully determined by the 
ratio of chord to diameter] and that, conversely, the ratio k is 
uniquely determined by the angle 0. In short, using modern 
terms, k is a function of 6. This function the Greeks called 
Xopdy. Translated as chorda into Latin, it was used as a standard 
term until superseded by the term sinus. 

When viewed in this light, the law of sines is an immediate 
corollary of the theorem that any three non-collinear points 
determine a unique circle. Consequently, the angles A, B, G 
of any triangle may be viewed as inscribed angles of some circle, 
and the corresponding sides, a, b, c, as chords of the same circle. 
If then we denote by D the diameter of that circle, we can 

a = D sin A, b = D sin B, c = D sin C, . (15. i) 

and this is the law of sines. (Figure 29$.) 

This interpretation shows that the law of sines is not an 
exclusive attribute of the triangle, and we shall presently see 
that Ptolemy was fully aware of this. Indeed, the preceding 
argument can be extended to any cyclic polygon, i.e., to any 
polygon the vertices of which lie on one and the same circle. 
A case in point is the cyclic quadrilateral which played quite an 
important part in the Ptolemaic treatment. 

We conclude that classical geometry contained all the elements 
required for the "solution 5 ' of triangles and rectilinear con- 
figurations generally. How about analytic trigonometry: the 
trigonometric functions of the sum or difference of two angles, 
of multiples and of half-angles! Well, not only were these pro- 
cedures known to Ptolemy, but they were used by him with 
telling effect. What is more, all these calculating media were 
derived from a single proposition which we call today Ptolemy's 
theorem, but which the Almagest modestly designated as the 



This is the theorem : if the four vertices of a quadrilateral are 
concycliC) then the sum of the products of the opposite sides is equal to 
the product of the diagonals of the quadrilateral. In Figure 30, 

AC-BD + BC- AD = ~AB CD. . (15.2) 

Even more remarkable than the consequences which Ptolemy 
drew from this property of cyclic quadrilaterals is the proof of 
the theorem. For here are displayed the same nimble virtu- 
osity, the same apparent artfulness which we encountered in 


Hero's proof of the formula for the area of a triangle, or in 
Euclid's proof of the hypotenuse theorem. This virtuosity suggests 
that the proposition did not spring from the brain of an 
astronomer, but was the discovery of some brilliant geometer, 
most likely Apollonius. 

As in the theorems just mentioned, it is not the proof that 
is hard to comprehend, but the "stratagem" behind the proof. 
The stratagem in this case is to determine a secant CX which 
partitions ABC into two triangles of which the first, ACX, is 
similar to CDS, and the second, BCX, is similar to CD A. This is 
accomplished by erecting an angle ACX equal to DCS. By the 
same token, angle BOX becomes equal to ACD. The rest of 
the proof is standard procedure : from the similitude of the 



two pairs of triangles we draw two proportions which, in turn, 
lead to the two relations, 


and CD - BX = BC AD. 

Adding these and remembering that AX + BX AB, we 
obtain the theorem sought. 


In applying the lemma to the calculation of chords, Ptolemy 
uses special quadrilaterals in which either a side or a diagonal 
coincides with a diameter of the circumscribed circle. Thus, in 
the case of the sum of two angles (Figure 310), the diameter 
AB is a diagonal. In what follows we have replaced the Alma- 
gest terms chord and complementary chord by sine and cosine. We 
have also assumed the diameter of the circle to be equal to i . 
Thus, one pair of opposite sides is made up of sin a and cos ft, 
the other pair of sin ft and cos a, while the second diagonal is 
sin (a -f- ft). A direct application of Ptolemy's theorem leads 
to the addition formula: 

sin (a + ft) = sin a cos ft + sin ft cos a 


In Figure 31^, the diameter of the circle is taken for one of 
the sides. Ptolemy's theorem then yields the difference formula 

sin (a ft) = sin a cos ft sin ft cos a . (15-4) 


Finally, Figure 32 shows how Ptolemy handled the dichot- 
omy problem; the chord of an angle being given, calculate the chord 
of the half-angle. Here the chords DB and 23C* are equal, their 



common length being sin |#; the chord BC is sin 6, and the 
diagonal AD is cos 6. Thus 

sin %6 + sin \6 cos 6 = cos \B sin 6 
This, in turn, leads to the dichotomy formula 

tan %0 cosec 6 cot 6 . . ( 1 5 . 5) 

which, incidentally, was used by Archimedes four hundred 
years earlier for the computation of n. 

These principles were applied in the Almagest to the computa- 
tion of a table of chords, which, from all accounts, was largely 
a reproduction of the one appended to the lost treatise of 
Hipparchus, a circumstance that casts added doubt on the 
authorship of the theorem which bears Ptolemy's name. The 
table is of great historical interest, not only because it was 
the first of its kind, but because its very conception was a 



radical departure from classical tradition. Besides, it offers an 
insight into the difficulties with which calculators of that 
period were beset. 

To bring these handicaps out in sharper relief I shall 
present here in outline Ptolemy's calculation of the sine of 
one degree. Step L The sine and cosine of 18 were taken directly 
from the golden section triangle. (See Chapter Five and Figure 
8.) Step II. The functions of 15 were derived from those of 
30 by dichotomy. Step III. The functions of 3 were derived 
from those of 18 and 15 by means of the difference formula 
(15.4). Step IV. The functions of i and of | were obtained 
from those of 3 by two consecutive dichotomies. Step V. Finally, 
the sine of i was derived from the sines of i and | by 

The latter was based on an inequality which had already 
been used by the astronomer Aristarchus, a contemporary of 
Archimedes. In fact, one commentator attributes it to Hippias 
of Elis, which is not so far fetched as it may first appear, 
inasmuch as it is, in a sense, suggested by the quadratrix. 
Indeed, the quadratrix is represented by the function #/tan x 
whereas the inequality back of Ptolemy's interpolation deals 
with the related function sin x/x. 

Figure 33 is the graph ofjy = sin x/x. Observe that the function 
is steadily decreasing as x varies from o to n. This means that if 
a, 8 and /? are three consecutive values of x, then 

sin a sin 6 sin ft 

from which we draw the inequality used by Ptolemy : 

f) ft 

-sin a > sin 6 > -3 sin /? . . (15*6) 

a ft 

In the second place, the function attains a maximum for x = o, 
which means that for small values of x, the curve is quite 


"flat." Thus, the interval between sin a/a and sin /?//? is small, 
if a and /? are small, and this is what accounts for the extra- 
ordinary precision of the method. 


We find, indeed, by applying the inequality to a = 90', 
6 ='6o' and /? = 45', that 


- sin 90' > sin 60' > ^ s in 45', 

3 3 

O' oi 745 . . . > sin i > 0*01745 

We are thus justified in taking sin i = 0-01745, which agrees 
with the value drawn from a five place table. It also agrees 
quite closely with the value given in the Almagest table: 
sin i = i' 3". 

Let me say here for the benefit of the puzzled reader that 
along with the Babylonian measures of time and angles, 
Hipparchus adopted the so-called sexigesimal fractions. Thus, 
i' 3" meant 1/60 + 3/3600. Ptolemy followed suit, and, as a 
matter of fact, this method of writing fractions was in common 
use even in the days of Copernicus. However cumbersome this 
notation may appear to us today, it was in keeping with the 
spirit of the time, and may indeed be viewed as a precursor 
of the decimal fraction which in its present form is barely three 
hundred years old.* 

And while on the subject of notation let me add that the 
terminology and symbols used in trigonometry today are 
largely those which Euler had introduced about two hundred 
years ago. The earlier nomenclature and notation were in 

* See this author's Number, the Language of Science, 4th edition, Allen & Unwin, 
1954. Pages 257 and 258. 




many respects as inarticulate and as awkward as those used 
by Ptolemy and his Arabic and Hindu successors. 


The essential difference between the classical geometry of the 
triangle, as Euclid taught it, and the trigonometry of Hip- 
parchus can be summed up in a single word, goniometry. Without 
goniometry, trigonometric ratios are but empty symbols and 
trigonometric laws fruitless formalities. The ratios are raised 
to the dignity of functions, and the laws acquire universal 
significance through a principle which co-ordinates angular 
measurement with the measurement of lengths. Such a principle, in 
turn, rests on the assumption that it is possible to establish a 
one-to-one correspondence between the arcs of a circle and the segments 
of a line. 

Classical geometry had no goniometry in this broad sense 
of the term. The Elements of Euclid defined congruent angles ; 
defined addition and subtraction of angles; the multiples of 
an angle ; dichotomy. It taught how to divide a complete 
revolution into 3, 5 and 15 parts, and how to subdivide any 
angle into z n parts. Still, all these definitions and construc- 
tions were strictly operational, and the operations were restricted 
to manipulations by straightedge and compass. To solve a triangle 
was a graphical problem which, generally, had no arithmetic 

Classical geometry had an angular unit: the right angle or 
quadrant. But only such fractions of the quadrant were recognized 
as bonafide angles as could be reached by cyclotomy; any others, 
such as 1/7, 1/9 or 1/90, were outside the "angular pale." 
And such was the power of the interdiction imposed by the 
"divine instruments" that even a Eudoxus or an Archimedes 
could not shake off these inhibitions. 

Then an astronomer rushed in where geometers had feared 
to tread. Not content with foisting on classical geometry an 
unwanted goniometry, he added insult to injury by defying 
the sacred rules of cyclotomy. Any goniometry which accepted 


the general angle at par with the "cyclotomic" was a radical 
departure from Greek tradition. Still, Hipparchus could have 
preserved a modicum of classical decorum by adopting for 
unit some angle within the scope of the divine instruments, such 
as 1/60, 1/80 or 1/96 of a quadrant, in lieu of the Babylonian 
1/90, which lay outside the "pale." It was a choice between a 
Greek schism and a Babylonian heresy. Hipparchus chose 



Not yet has man learned how to celebrate his highest 


What is the scope of the Greek contribution? What place 
should the historian assign to the Greeks in the evolution of 
mathematical thought and technique? Is the statement "in 
mathematics all roads lead back to Hellas" a just appraisal, or 
just a specious metaphor? Was it devotion to Greek principles 
that brought about the prodigious progress of the last few 
centuries, or was it renunciation of Greek mathematical 

As I ponder over these questions, there come to my mind 
the words of Poincare': "To doubt all or all believe are two 
equally convenient solutions, in that both dispense with 

My account of the Greek Bequest touched on many problems 
and issues, which had their reverberations in modern times; 
and yet, it is not the whole story. A number of equally impor- 
tant achievements had to be relegated to the next volume of 
this trilogy. Lack of space was one reason for this deferment. 
The other was my feeling that these issues could be presented 
in sharper relief when etched against the background of the 
time and the place where they had attained fruition. 

Among these topics will be such significant achievements 
as the quadratures of Archimedes which anticipated the integral 

1 80 


calculus of Newton by two thousand years; the Conies of Apol- 
lo nius which foreshadowed the analytic geometry of Descartes; 
the porisms of Euclid and Pappus which contained the germs 
of the projective geometry of Desargues; the Arithmetica of Dio- 
phantus which inspired the number-theoretical discoveries of 
Fermat; the principle of exhaustion; the postulate of parallels; the 
curves of Diocles and Nicomedes; Euclid's brilliant studies in 
prime and perfect numbers. 

An imposing array, yet all parts of the mathematical heritage 
bequeathed to us by the Greeks. If these topics be ranged 
alongside of those recorded in the present volume, the list 
would read like a table of contents of an encyclopedia of 
modern mathematics. By the same token, the words "all roads 
lead back to Greece" would resound like an apt description 
of the relation between classical and modern mathematics. 

When, however, we examine closer the twelve odd centuries 
which separate the end of the classical from the beginning of 
the modern era, we find this metaphoric appraisal specious, 
to say the least. The first eight centuries of that period are 
known as the Dark Ages. There has been a tendency in recent 
years to avoid this term, but I know of no more fitting epithet 
unless it be interminable night. There were no roads leading to 
Hellas in that barren wasteland. Just trails littered with the 
ruins of Greek culture. Occasionally an errant monk would 
stalk among the ruins in vain search of a Greek rationale 
which would vindicate his wild obsession. And that was all. 

When at long last the obsession had run its course, there 
came to Western Europe a magnificent upsurge known as the 
Renaissance. Pent-up energy released found creative outlets in 
the arts, in letters, in music and philosophy. However, there was no 
corresponding upsurge in mathematics at that time. Feeble 
attempts were made to revive Greek classics through Latin 
translations of their Arabic versions, but these efforts had no 
lasting effect. Indeed, the only mathematician of the period 
worthy of the name was Fibonacci, and he was more interested 



in spreading Arabic ideas than in restoring the glory of 

Then came the seventeenth century and an upsurge which, for 
intensity and extent, had no equal in the annals of mathe- 
matics. Most of the subjects which grace the mathematical 
curriculum of a modern college came into being during that 
period. Geometry, analytical, projective, infinitesimal; theory of equa- 
tions and number theory; the calculus, theory of functions, infinite 
series, theory of curves; probabilities, theoretical and celestial mechanics. 

Now, all this happened in less than one hundred years. 
Indeed, while the Isagogs of Vieta, which ushered in the new 
era, had appeared in 1592, the full significance of his symbolism 
was not realized until after the author's mysterious death in 
1603. On the other hand, in 1686, when Newton's Principia 
Mathematica was published, the new mathematics was already 
a fait accompli. Thus, measured by contemporary standards, 
there was more ground covered in this, the first, century of 
modern mathematics than during the nearly one thousand 
years which separate Thales from Pappus. Why? 

We certainly cannot ascribe this prodigious progress to the 
superior ability of the mathematicians of the seventeenth 
century, or to their deeper insight. The phenomenal virtuosity 
of the Greek masters and their almost uncanny intuition 
speak for themselves. Indeed, so circumscribed were the 
methods of the classical era, that almost any new problem 
called for a tour de force which only a virtuoso could perform. 
Again, to say that Fermat, Descartes or Newton had tools 
available which were not known to the Greeks is begging the 
question, inasmuch as these tools were forged by the very 
men who used them. What were the stimuli that urged these 
men on, stimuli which, apparently, did not exist in the Greek 



What classical mathematics needed to become an organized 
whole, and to qualify, at the same time, as spokesman of the 
sciences which lean on it for counsel and approval, was not 
more or greater men of genius, and not new principles or 
concepts: it needed a new language. Does this mean that the 
Latin of Descartes and Newton was better adapted to express 
mathematical thought than the Greek of Apollonius or Pappus? 
No, indeed! The question as to which of the sundry languages 
evolved by civilization could best serve the needs of mathe- 
matics is of no historical significance, since the verdict of 
history has been that no such language can ever fill the exacting 
demands of mathematics. 

As I write these lines, I think of the cynical phrase of 
Talleyrand: "Speech has been given to man that he may 
disguise his thoughts." A choleric historian might paraphrase 
this into: "Speech has been inflicted on man to obscure thought, 
paralyse action and impede progress" and could invoke the story 
of classical mathematics to confirm his words. A more benign 
interpreter of human history would counter: "Language is 
not a gift of Providence. It grew out of the needs of a social 
being to convey to other such beings his wishes, entreaties and 
commands ; to share with other such beings his hopes and 
fears; to implore, placate, cajole and exorcise the mystic forces 
which controlled his destiny. It antedates inference and deduc- 
tion by countless eons ; it bears the indelible imprints of the 
chaotic mist through which human intuition was groping, 
ages before the advent of reason." 

Thales of Miletus armed intuition with a brain, and out of 
the nebulous mist emerged mathematics. But neither Thales 
nor his followers armed the thinker with an organ of speech 
which would fittingly express his thoughts, subtle yet precise, 
or describe the countless forms which his imagination could 


conjure up. Greek mathematics had to depend on common 
speech, a medium replete with ambiguities, yet inflexible; 
open to inconsistencies which it could not detect; where an 
interchange of words could jeopardize meaning, and where 
emphasis could be attained only through intonation. These 
were the handicaps under which Greek mathematics laboured 
throughout the thousand years of its existence. 

And then, as though by magic, mathematics was freed from 
the vagaries of human speech and presented with a language 
all its own. I use the word magic advisedly, for, the most 
striking feature of the event was the spontaneity and rapidity 
of this transition from the old mathematics to the new. It 
began at the threshold of the century, and by 1650 the new 
medium had already infiltrated into practically every field of 
mathematics, pure or applied. 

Whenever I reflect on that epical transition, there comes 
to my mind the legend of Hesiod, according to which Zeus 
had swallowed his wife Metis when she was pregnant with 
Athena, acting on a warning that his children by her might 
prove stronger than himself and dethrone him. But Prometheus 
split open the head of Zeus, and Athena sprang forth fully 
armed and uttering a loud shout of victory. Prometheus was 
not aware of the enormous potentialities of the liberated 
goddess. And neither was Vieta, the Prometheus of Mathe- 
matics, aware of the revolutionary outcome of his discovery. 
Indeed, had he been so aware, he would have hailed the 
discovery as lingua mathematica, instead of christening it with 
the vapid name of logistica speciosa. 

"Not yet has man learned how to celebrate his highest attain- 
ments." How fittingly these words of Nietzsche apply to 
the history of mathematics! The two epoch-making events in 
that history were the principle of deductive reasoning inaugurated 
by Thales and the symbolic algebra of Vieta. Yet, not only do 
we not commemorate these events, and not only have the 
men to whom we owe these achievements been all but for- 



gotten, but we even lack names to identify these achievements 
for what they are. 

Indeed, few of our textbooks mention the Vieta discovery 
at all, and those that do, identify it as literal notation, a term 
which is not only pointless, but actually deceptive, since the 
use of the letters of an alphabet for symbols is wholly irrelevant, 
and the idiom is certainly more than a mere notation. In 
fact, even the term language does not adequately describe the 
scope and power of the medium. For, while the idiom will 
discharge the principal functions of any written language, it 
will also perform feats which even the most eloquent orator 
or the most inspired poet cannot hope to attain. 

Thus, the change of structure in a sentence will, as a rule, 
destroy the intended meaning, or, at best, produce a pun, even 
if the rules of grammar have been strictly obeyed. On the other 
hand, in a symbolic relation a transformation which conforms 
to the laws of algebra reveals the equivalence of two forms, 
and this means a theorem in the field which has marshalled 
these symbols to its aid. 

Again, when words in a sentence are given a new connota- 
tion, the result is an ambiguity, or, at best, a metaphor. But if a 
symbolic relation is valid in two distinct fields, then any 
consequence of the relation in one of the fields has its counter- 
part in the other. Such isomorphisms have led to many valuable 
discoveries in applied mathematics. 

Finally, if the words of a sentence are subject to certain 
reservations, then to disregard these restrictions means to 
risk grave consequences; indeed, we owe most of our errors 
to just such abuse. On the other hand, a symbolic relation 
in which the entities involved have originally been restricted 
in type or range will often retain its meaning when the restriction is 
removed and thus lead to fruitful generalizations. The evolution of 
the number concept from integer to vector is a case in point. 

This, then, is the historical significance of Vieta's discovery : 
it not only endowed mathematics with a language, but armed 
it with such powerful weapons as paraphrase, analogy, generaliza- 
tion. Thus did Vieta turn a tongue-tied thinker into a fluent 
and convincing speaker, and, at the same time, immensely 
enriched the thinker's creative and critical faculties. 


The publication in 1630 of Galileo's Dialogues on the Two New 
Sciences provided a second powerful stimulus to the mathe- 
matics of the seventeenth century. In that work Galileo sought 
to extend the notion of velocity to non-uniform motion. He 
inquired : what would happen to the average speed of a moving 
particle when the time-interval was gradually reduced? This 
led him to the concept of instantaneous velocity. The latter he 
defined as the limit towards which the average speed tends 
when the time-interval diminished indefinitely,, which, of course, 
was in all but name the derivative of space with respect to time. 

It was another case of an astronomer rushing in where 
geometers feared to tread. The spell was broken: Fermat was 
quick to adopt the new ideas to geometry; a veritable orgy of 
applications followed, and infinitesimal analysis was born. 
Infinite processes, infinitesimals, limits concepts which Greek 
geometers had shunned with a circumspection akin to awe 
were henceforth to be used as legitimate instruments of mathe- 
matical reasoning and technique. Thus did Galileo inadvert- 
ently cut the Gordian knot of a taboo which had plagued 
mathematics since its inception. 

Still, the enormous success of infinitesimal analysis should 
not blind the historian to the fact that the discipline owed its 
existence not to a new conception, but to a bold overt break 
with an age-old tradition. We should also remember that 
Greek mathematical history is studded with attempts to shake 
off the taboos imposed by that tradition ; that these attempts, 
timid and disguised at first, were becoming more resolute as 
mathematics advanced; and that the Collection of Pappus is 
interspersed with problems where the infinite with its sundry 
ramifications was used directly, even if not explicitly. 

The Greek title of the Pappus work was Synagogs, which stood 
for collection or assembly. Quite a fitting title, too, for, here 



Pappus assembled a veritable pageant of classical problems, 
ranging from the earliest days of Greek geometry to his own 
times. Some were old questions in a new guise, others were, 
undoubtedly, due to Pappus himself, but all bear the elegant 
touch of a master mathematician and skilful teacher. Indeed, 
the very style and arrangement of the material suggest that 
he had been the head of a school of high standing. 

The historical significance of the Synagogs is that it serves as 
the best available source on the scope and ken of those mathe- 
matical classics which had perished during the Dark "Ages, 
and, at the same time, sheds light on the state of mathematics 
at the eve of the blackout. But it does more than that. For, by 
revealing the trends of mathematical thought and technique 
at the close of the classical era, it brings out in true perspective 
the transition from the old mathematics to the new. 

The juxtaposition of the two works, the Synagogs of Pappus, 
that swan song of classical mathematics, and the Isagogs of 
Vieta, the first significant book of the modern era, would 
disclose the origin of many current ideas, and would, at the 
same time, clear up many perplexing historical questions. 
This is the reason why I have deferred the study of the closing 
era of Greek mathematics to the next volume of this trilogy. 
Still, the era is a part of the classical panorama: hence, a 
fitting climax to a story of the Greek Bequest. And so I shall 
close my narrative with a retrospective glance at that crucial 
era in the history of man, when the fate of thought was hanging 
in the balance. 


History may be likened to one of those tapestries of intricate 
weave in which an eye can discern almost any design con- 
ceived in advance, any one at variance with any other, yet 
none conflicting with the whole. 

One might picture the death scene of Hellenism as a host 
of saints chanting Hosannas on the grave of a pagan who had 
died in despair and decay from hypertrophy of the mind and 
atrophy of the soul. Or, one might picture it as a horde of 


fanatics massed under the banner of ratio delenda est, glaring 
with glee at a prostrate Prometheus, felled at the prime of life. 
Between these extremes one might intercalate a variety of other 
designs, no two agreeing with each other, yet, every one drawing 
aid and comfort from the confusing accounts of an age of 
much heat and no light. 

A bewildering freedom of choice, but, actually, no freedom, 
since one's choice is predicated by one's temperament, up- 
bringing and entourage. Nor is the historian free from such 
bias, and I, for one, have never concealed my own. Homo sum et 
humanum nihil a me alienum puto. 

Many centuries lie between us and those tragic days, but 
the ideological conflict between the Hellenists and the apologists 
has not abated. The one speaks of the event as of darkness 
engulfing light and regards modern progress as the reincarna- 
tion of Hellenism; the other describes the end of Greek culture 
in terms of degeneracy, decadence, decline and decay. 

I do not feel competent to weigh the merits of the contro- 
versy in fields other than my own. However, I did make an 
honest effort to appraise the mathematics of that era. I read 
and re-read the Metrica of Hero, the Almagest of Ptolemy, the 
Arithmetica of Diophantus, the Collection of Pappus, and sundry 
other works which had escaped the vigilance of the saints. 
I found there a growing understanding of the issues which had 
stalled their mathematical predecessors, and a groping for 
means to resolve these issues. I found pride of achievement, 
and prouder yet visions of conquests to come. But nowhere 
did I detect evidence of decay or traces of decline. Yes, I, too, 
have scanned that intricate weave to discern the stage on which 
had been enacted the last hours of classical mathematics, and 
here is what I saw from where I stood: 

A bright day was shining, lofty vistas were looming, fresh 
breezes were scattering the cobwebs of ancient taboos, when 
the lights went out, and the curtain of history dropped on the 
Grand Drama of Hellas. 

1 88 


Alexandria, 35 

Algebra, 92, 131, 132, 144, 185 

Algerithm of 

Euclid, Chapter 12, 152, 153, 159 

Hero, 155, 159 

Lagrange, 167 

Wallis, 147, 148, 150 
Almagest, 160, 170 
Analysis, 92 
Anaxagoras, 39 

Angular measurement, 1 70, 1 78, 1 79 
Antinomy, 103 
Apollonius, 17, 34, 35, 88, 173, 181, 


Arc, 96, 128 
Archimedes, 92, 124, 141, 148, 

Chapter 13 

Archytas, 37, 38, 39, 80 
Area, 100, 160 
Aristarchus, 31, 176 
Aristophanes, 78 
Aristotle, 27, 31, 89, 122, 125 
Astrology, 69 

Astronomy, 169, 170, 176, 178 
Athena, 184 
Azimuth, 35, 51 


Babylon, 29, 43, 44, 169 
Blavatzky, 67 
Bombelli, 151 
Brahmagupta, 164, 165 

Calculus, 181, 182, 186 

Cantor, Georg, 103 

Cardano, 17 

Cassini, 38 

Cavalieri, 17 

Character of number, 41 

Cheops, 46 

Chord, Chapter 15 

Circle, 91, 161, 162 

Circular segment, 128 

Claussen, 136, 137 

Comte, 19 4 

Condorcet, 72, 79-81 

Conic sections, 35, 36, 88, 142, 143 

Conjugate, 167 

Continued fraction, 50, Chapters 12 

and 13 
Copernicus, 31, 67, 177 

Crescent, Chapter 10 

Crotona, 23 

Cubic curves, 88 

Cubic equation, 44 

Cuneiform, 43 

Curvature, 86 

Curve, 128, 142, 143, 144 

Cycle, 150 

Cyclotomy, 64, 136, 178 

D'Alembert, 46, 50 

Dante, 17 

Decagon, 62 

Decimal fraction, 151 

Dedekind, 103 

Degree, 142 

Delian problem, 38, 39, 41, 122 

Democritus, 25 

De Morgan, 39, 75 

Desargues, 181 

Descartes, 141, 181, 182, 183 

Dichotomy, 140, 175, 178 

Digital, 118 

Dinostratus, 138 

Diocles, 181 

Diogenes Laertius, 23 

Diophantus, 110, in, 181 

Dirichlet, 167 

Distance, 96 

Divine proportion, 56, 61, 65 

Dodecahedron, 64, 65, 66 

Dorodnov, 136, 137 


Eccentricity, 68 
Ecliptic, 53 

Egypt, 22, 29, 43, 46, 47 
Einstein, 76, 82 
Elements : see Euclid 
Ellipse, 38 
Equiareal, 100-103 
Eratosthenes, 17, 34, 38, 39, 41, 121, 


Error, 149 
Escribed circle, 162 
Euclid, 17, 24, 34-8, 47, 87, 89, 90, 95, 

97-1 1 1, 153, 181 
Eudemus, 24, 25 
Eudoxus, 37, 39, 86, 97, 153, 178 
Euler, 64, 113, 117, 120, 151, 168 
Euripides, 38 
Extreme and mean reason: see Golden 



T UJf lilJb 

Famous problems, 81, 92 

Fechner, 57 

Format, 72, 76, 96, 117, 120, 136, 141, 

165, 181, 182 
Fibonacci, 113, 114, 119, 148, 150, 165, 

181, 182 
Foursquare theorem, 108 

Galileo, 17, 67, 186 

Galois, 15, 19, 151 

Gauss, 32, 109, 117, 151, 167 

Genus, 142 

Geodesic, 86 

Geometry, Chapter 3 


Analytic, 30, 141, 142 

Deductive, 41, 44, 45 

Empirical, 41 

Non-Euclidean, 73, 96 
Giza, 46, 59 

Golden section, 50, Chapter 5 
Goniometry, 124, 178, 179 

Hero of Alexandria, 34, 148, 188, 

Chapter 14 
Hesiod, 184 
Heuristic, 59 
Hieronimus, 50 
Hilbert, 84, 109 
Hipparchus, Chapter 15 
Hippias, 25, 39, Chapter 1 1 
Hippocrates of Chios, 25, 39, 41, 42, 

Chapter 10 

Hippocrates, the physician, 121 
Homogeneous, no 
Horoscope, 69 
Huygens, 148, 149, 151 
Hypotenuse theorem, 28-32, 48, 59, 

Chapter 8 
Hysteresis, 78 

Icosahedron, 60, 105 

Ideal, 73 

Incommensurable, 95, 103 

Infinite process, 60, 105, 128, 140, 186 

Interpolation, 157, 176 

Isagogs, 182, 187 

Isomorphism, 185 

Iteration, 156 

Jacoby, 151 


Kepler, 55,66-71 
Klein, 95 

Lagrange, 113, 117, 151, 166 

Landau, 136, 137 

Language, 183-5 

Law of sines, 171, 172 

Legendre, 117, 151 

Leibnitz, 60, 142, 143 

Leonardo da Vinci, 17, 56, 60, 65 

Leonardo of Pisa, 1 1 3 

Limit, 153, 1 86 

Linkwork, 91 

Liouville, 151 

Logistica, 108 

Logistica speciosa, 184 

Lune, 122 


Mean proportional, 131 
Mechanics, 87 

Menaechmus, 37, 38, 39, 138 
Menelaus, 169, 170 
Meniscus, 123 
Mersenne, 117 
Metampsychosis, 23 
Metis, 184 
Metric, 95 
Michelangelo, 17 
Miniature, 49 
Music of the spheres, 69 
Mysterium Cosmographicum, 67, 68, 

fin *rn 


Newton, 17, 32, 70, 142, 181, 182, 183 
Nicomedes, 181 
Nietzsche, 180, 184 

algebraic, 143 

complex, 96 

irrational, 58, 95, 102, 148, 153 

perfect, 181 

prime, 120, 181 

rational, 104, 105 

transcendental, 73 
Number e, 73 

Number 77, 41, 50, 141, 143 
Number theory, 108, 182 

Occult, Chapter 5 
Orbit, 67 




Paccioli, 6 1, 65 

Pappus, 24, 25, 34, 36, 138, 181, 1 88 

Parabola, 124, 153 

Parallels, 72, 73, 82, 96 

Parameter, 112, 113, 164 

Parity, 115 

Pell, 1 66 

Pentacle, 63 

Pentagon, 62 

Pentagram, 62 

Period, 150, 151 

Phoenicians, 18, 20, 21 

Plato, 25, 26, 55, 65, 86, 87 

Platonic solids, 65 

Platonists, 63, 1 12 

Plutarch, 86, 87 

Poincare", 45, 103, 180 

Polyhedron, 64 

Polynomial, 142, 143 

Porism, 36, 37 

Positional numeration, 113 

Postulate, 90 j 

Primitive, no 

Prometheus, 184, 188 

Pseudomath, Chapter 6 

Ptolemy, 160, 170, 172, 188 

Pyramid, Chapter 4 

Pythagoras, Chapters 2 and 3 


Quadrable, 124 

Quadrant, 178 

Quadratic equation, 55, 131 

Quadratic surds, 49, 52, Chapter 13 

Quadratrix, 88, Chapter n 

Quadrature, 41, 138, 180 

Quadrilateral, 172 

Rational approximation, 

Chapters 12 and 13 
Rational triangle, 104 
Rectification, 122 
Reguia falsi, 157 
Relativity theory, 75, 96 
Rome, 1 6, 17 

Schopenhauer, 97, 169 
Sexagesimal, 170, 177 

49. 59> 

Shadow, 47, 50-4 

Similitude, 52-4, 97-9 

Smith, James, 77, 81 

Solon, 22 

Space, 86, 96 

Spectrum, 60, 146, 151 

Spencer, 19 

Spengler, 19 

Spherics, 170 

Spinoza, 8 

Square, 58 

Square root, 150, 152, 156, 168, 

Chapter 13 
Squaring the circle, 31, 40, 41, 74-80, 


Steiner, 138 
Straight-edge compass, 40, 41, 88, 89, 

124, 178, Chapter 7 
Straight line, 90 
Sylvester, 121 
Symbol, 185 
Syracuse, 21 

Taine, 19 

Talleyrand, 183 

Tchebotarev, 136 

Thales, 34, 42, Chapters 2 and 3 

Timaeus, 55 

Transcendental, 142, 143 

Trapezoid, 132 

Trigonometry, Chapter 15 

Triples, Chapters 9 and 14 

Trisection, 73 

Two square, 117 


Vieta, 17, 113, 182, 184, 185, 187 


Wallis, 146-151, 166 
Weierstrass, 103 
Whewell, 34 
Wolfskehl, 76, 77 

Zeno, 103 
Zeus, 184 

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